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Large, precision optics can now be manufactured with surface figures specified at the sub-nanometer level. However, coatings and gravity deform large optics, and there are limits to what can be corrected by clever compensation. Instead, deformations caused by stress from optical mounts and deposited coatings must be incorporated into the optical design. We demonstrate compensation of coating stress on a 370mm substrate to λ/200 by a process of coating and annealing. We also model the same process and identify the leading effects that must be anticipated in fabrication of optics for future gravitational wave detectors and other applications of large, precisely figured optics, and identify the limitations inherent in using coatings to compensate for these deformations.
© 2015 Optical Society of America
1. Introduction
Recent advances in many fields have raised the stakes in optical fabrication, pushing the boundaries of even the best fabrication and coating techniques. Gravitational wave detection, for example, requires optical elements that are among the most difficult to fabricate, and creating them has long pushed the boundaries of optics fabrication, coatings and metrology [1–3, 5, 6]. Detectors now under design and construction demand sub-nm level surface figure specification. Large UV and X-ray optics, due to the short wavelengths involved and the grazing incidence conditions of their use, also require extreme measures to meet specifications [7]. Astronomical instruments such as large aperture spectrometers [8] also raise new challenges. High-power pulsed laser systems place additional demands on optics that require unusual coatings and a thorough understanding of heat loads [9]. Finally, the metrology standing behind this fabrication will inevitably demand similarly exquisite optics as reference surfaces.
Optical substrates usually experience distortion when thin film coatings are applied. This is mainly due to the compressive or tensile stresses in the coating materials due to the deposition process, and also the differential thermal expansion between the coating and substrate materials. The exact stress in the coating is difficult to predict accurately since coatings are usually composed of many layers with varying intrinsic stress, and thermal gradients are inevitably present in the coating and substrate throughout the coating process. They also depend sensitively on the details of the coating process and geometry. In addition, differential thermal expansion can be significant if the optics are cooled to cryogenic temperatures, which has been suggested for some gravitational wave detectors, particularly the Japanese Kamioka Gravitational Wave Detector (KAGRA) [2]. Crystalline coating materials that are being considered for use in future detectors [3] will make these distortions anisotropic due to different mechanical properties along and across the crystal axis, adding further complexity to the problem.
Optics also distort under their own weight [4], and when the surface figure specifications reach the sub-nm level, this requires that the optics be figured so that they take the desired shape when mounted in a particular way. In present gravitational wave interferometers, the main optical components are always vertically oriented and suspended to isolate them from their seismic environment. Weight will distort the optics, and this must be taken into account when figuring them.
We can approach this problem in two ways. First, we may perform surface figure measurements with the optics suspended just as they will be when placed in the interferometer. Unfortunately, this is likely to raise the cost of the metrology substantially, since it demands that the mount be reproduced in the optics shop for testing during the figuring and coating process. In the case of suspension from blocks bonded to the side of the optics [10]– as shown at left in Fig. 2 – this much more cumbersome, since it would require surface figure measurements to be made during fabrication with the blocks already bonded, to evaluate the surface figure under gravitational sag. In addition, some of the next generation of gravitational wave detectors may operate cryogenically, adding stress due to differential thermal expansion of the coating materials compared to the substrate. Measuring surface figures under cryogenic conditions repeatedly during fabrication would be prohibitively expensive.
Our second alternative is to perform our surface figure measurements in a well known configuration where we understand the gravitational deformation well, and use a combination of finite element modeling and empirical tests to estimate the expected deformation the optic will experience when mounted. So far, the optical tolerances have been loose enough that this type of modeling has primarily been used to verify that the expected deformation from the mounts lies below the specification tolerance. However, we anticipate that future interferometers will require optics specified at the sub-nm level, requiring that these deformations be predicted accurately and accounted for during the substrate fabrication.
In this work, we develop a finite element model for coating stress and its compensation. We compensate for the coating stress distortion with a counter-coating on the opposite side of the optic. We performed an iterative process of deposition and annealing to accomplish this. In the end, a coating distortion of several hundred nm is reduced to < 2nm across the clear aperture of 220 mm diameter. We also find that the coating thickness uniformity in this process was better than 0.2% across the clear aperture. Finally, we investigate the distortions due to weight and their effect on fabrication of optics at the sub-nm level.
2. Distortion model
We have constructed a model for simulation of stress induced deformation in the optical substrates under study here. We model the system typically used in our metrology for large optics, consisting of a coated or uncoated substrate, lying horizontally, supported by foam material to minimise distortions imposed by the support.
Since we expect both the coating and the substrate to be radially symmetric, it is sufficient to use a 2-d, axisymmetric model. The fused silica substrate dimensions (corresponding to the test substrate used in section 3) are 370mm diameter, 60mm thickness, 3mm chamfer, and the material properties are shown in table 1. The model also incorporates the open-cell foam material that supports the substrate.
Table 1. Material properties of fused silica used in the simulations for coating stress.
The foam material used in the measurements is Bergad 6083 polyurethane ”memory foam”, which provides extremely uniform pressure on the bottom surface [12]. Under these conditions, the stiffness of the substrate is over a million times that of the foam material.
Imposing gravity on the system results in compression of the foam material and a radially varying restoring force on the bottom of the substrate. In Fig. 1, we plot the results of this simulation, showing the deformation induced by the foam support. As shown, the substrate deforms by less than 0.2 nm in surface figure and thickness.
Fig. 2 Inertial test mass with ears, mounted vertically, showing predicted surface deformation (dimensions are in meters; greatest excursions are ∼ ±1.9 nm)
For calculations we used COMSOL Multiphysics. [11]. We have verified the accuracy of the model by comparing distortions of substrates supported at the edge, which sag substantially under their own weight. When compared with measured sag, the model agrees to within 10% with both both model calculations and measurements described in [12, 13]. When mounted vertically (Fig. 2), weight distortions are not radially symmetric and must be accounted for if these optics are fabricated at the sub-nm level.
To simplify and speed up the FEM convergence, we neglect gravity in the coating stress calculations. The stress imposed on the substrate by the restoring force of the foam is independent of any coating stresses, so we can safely assume that the gravity-induced deformation will be the same with or without coatings. We next add to the model a 5μm thick coating with the same material properties as the rest of the substrate, but with an additional compressive stress. The coating thickness is approximate and may differ from the actual thickness by as much as 20%. Figs. 3–4 show the resulting deformation for the coating geometry.
Fig. 3 Results of FEM calculation. The coating is uniform across the top surface and linearly tapers in thickness along the chamfer. The distortion is exaggerated for clarity.
Fig. 4 Results of FEM calculation, using a 5μm thick coating with stress = −325.5 MPa. Top: deformation of substrate by coating-induced stress. Middle: deformation with parabolic term removed, showing that the deformation is very nearly parabolic, but slightly different on top and bottom. Bottom: variation in thickness of the substrate. The curve fits well for 93 < r < 140 mm to an exponential decay with constant 16.1 mm. At r = 93 mm, the thickness varies by less than 0.035 nm. The green curves are shifted for clarity.
We have re-run these simulations with different coating stress and thickness, and the model predicts a deformation that varies strictly linearly in both, up to three times the values that correspond to our actual coatings.
One method for removing coating stress-induced deformation on optics is to apply a coating on the opposite side which exerts a compensating stress on the substrate, thereby reducing the deformation. To model the process of counter-coating, we apply a symmetry plane through the middle of the optic, parallel to the flats. The model predicts a deformation as shown by the blue curve in Fig. 5.
Fig. 5 Thickness change due to identical coatings placed on opposite faces of the substrate. The solid red curve shows the change in thickness calculated for one side coated, and the red dashed curve is twice this change.
As expected, the counter-coating eliminates the parabolic part of the deformation completely, leaving the central region extremely flat. However, as we approach the outside edge, the surface deviates significantly from flat. This ’edge effect’ becomes important where the distance to the edge is less than the substrate thickess.
The red curves in Fig. 5 show that the change in thickness due to the coating on one side is just half of the change in thickness of the final substrate. Therefore, in order to predict the overall change in surface figure for a counter-coated substrate, we need only measure the change in thickness after one coating. It is also clear that while counter-coating negates the parabolic distortion, thickness deformation is doubled for the counter coating.
We expect from these simulations that the counter-coating method will work well in the middle of the optics, but will fail to correct distortions at the outside of the optic. It would seem that these distortions must be anticipated, and the substrate must be pre-figured accordingly, to ensure that the balanced coatings will finally produce the desired figure to below 1nm, even on one side.
3. Experimental results
We have carried out the counter-coating process in detail with a test substrate. It was initially ion beam figured to within a few nm of flat. We measured the surface figure of the substrate before coating and between all stages of the coating process with CSIRO’s Large Aperture Digital Interferometer [14]. This instrument has a clear aperture of 320 mm and has an absolute accuracy of approximately 2.5 nm, and a precision of <1nm across the aperture, allowing differences in surface figure between measurements to be determined to below 1 nm.
The substrate was coated on one side with a multilayer dielectric, highly reflective (HR) coating of standard quarter-wave design. The coating was composed of alternating layers of tantala and silica, with a total thickness of approximately 5μm, deposited using the ion beam sputtering technique. On the second side, compensating layers of silica were deposited after which a thin 0.5μm antireflection (AR) coating was applied. We estimate the overall coating thickness to be approximately 5μm thick.
In Fig. 6, we compare the deformation due to the HR coating with the model prediction. Here we have used one free parameter, which is the coating stress. Comparison with the measured deformation and holding the thickness at 5μm required a stress of −325.5 MPa for the best fit between model and data for the top side in the middle of the substrate. The model deviates from the measurement as we approach the edge, and in this case we find more distortion than the model predicted.
Fig. 6 Top, comparison of predicted and measured stress-induced deformation due to first HR coating. Bottom, difference between model and measurement for both sides.
The same model also predicts the deformation of as yet uncoated bottom side, which was not coated. From the model we have good reason to expect the power term in the middle of the substrate to be identical on top and bottom as shown by the lack of change in thickness due to the first coating, (Fig. 4c). We conclude therefore that the difference in power terms on the two sides is primarily due to coating thickness nonuniformity. Due to the way our coatings are applied, we expect them to be rotationally uniform but to have some radial variation. The coatings are designed to optimally provide the reflectivity and absorption specifications required for gravitational wave interferometers (at the < 1ppm level), but it is still possible to meet these specifications with a small overall thickness variation. In this case, the implied thickness varies by approximately 0.2% of the overall coating thickness.
We find from experience that the reflectivity and absorption characteristics tend to change, and that the absorption loss tends to diminish, upon annealing at high temperature after the coating is applied. Ion beam sputtered coatings generally also have high compressive stress in the coating which is partially alleviated by subsequent annealing. For this reason, it is essential to anneal the substrate after coating, and it must be re-measured before the final AR coating is applied to the second side.
Figure 7 presents a summary of the changes in overall curvature to the substrate as the annealing and coating process proceeded. Here we plot the power terms for top and bottom sides as measured after each step in the process. Since an extra day is required each time we measure both sides, we only measured the top surface in a number of these steps.
Fig. 7 Summary of changes in power term due to coating and annealing. The measurements are ordered in time sequence from top to bottom. At some stages only side 1 was measured.
After the first annealing step, the net HR coating stress is reduced by nearly half. The following three silica coatings on side 2 are well matched to the refractive index of the substrate, and their primary purpose is to compensate for the coating stress on the HR side. The optically active layers of the AR coating on side 2 are much thinner, so these layers are required to achieve the final balance needed to bring the substrate back to flatness. To build up the required stress on side 2, we iteratively coated and annealed the substrate to empirically tune the thickness and stress that would bring the substrate back to flatness. Finally, the AR coating was applied, and the optic was returned to within less than 2 nm of flatness (in the Zernike power term, within the clear aperture of 220 mm.)
With the entire coating process complete, we compare again the surface figure changes with the model. In Fig. 8 we plot the overall change in the top surface figure from start to finish, in red. This curve reveals a small residual power term, showing that the final compensation was not perfect. By subtracting the parabolic component of this within the inner 120mm (slightly greater than the 220mm diameter clear aperture in this case), we arrive at the cyan curve. The green curve plots the model calculation as described in section above, showing good agreement out to about 125mm, but with substantial departure as we approach the edge of the substrate. Further FEM calculations have shown that the precise nature of the distortion near the edge depends sensitively on how the coating is distributed near the edge, and the extent to which the coating extends over the chamfer, details that our measurements of this substrate were not sufficient to reveal.
Fig. 8 Comparison between model and measured overall deformation of side 1 surface. The red curve shows the difference between the first and last measurements from Fig. 7. The cyan curve shows good agreement with the model (green) when the power term is removed from the experimental data.
4. Conclusion
The standard method of producing large, precision optics has been to figure the substrate to the final shape, and then apply surface coatings in such a way that the optic deforms less than the specification tolerance. Future production of optics for gravitational wave detection will require a combination of accurate modeling and empirical measurements of distortions caused by coatings and mounts. While ion beam figuring (IBF) can achieve sub-nm level figuring, this is only useful if the distortions imposed on the substrate after figuring is complete (coating stress, weight) are anticipated correctly. Carefully balancing the coating stress on both sides of a substrate is an essential part of this process, but edge effects make this method of limited value on thick substrates. Further work, including modeling and testing, will be required to understand the edge effects well enough to reliably predict them.
If we can reliably predict the weight and coating distortions, then it becomes possible to figure optics that, once coated, are flat on one or both sides, over the entire surface, down to the sub-nm level. We have shown that achieving this will require anticipation of the details of mounting, and thermal environment of the optic before the figuring process is complete. Thermal gradients within the substrate, driven by absorption of cavity light in the coatings, will inevitably create further distortions that must also be anticipated, and any compensation techniques may fall prey to similar limitations as those we have uncovered in this work.
Acknowledgments
We acknowledge assistance from Katie Green and Jeff Brown in the coating and measurement process, and we thank GariLynn Billingsley of Caltech for helpful discussions on the measurements. This work was supported by CSIRO, University of Tokyo, and California Institute of Technology through fabrication work for KAGRA and Advanced LIGO respectively.
References and links
1. The LIGO Scientific Collaboration, “Advanced LIGO,” Class. Quantum Grav. 32, 074001 (2015). [CrossRef]
2. K. Somiya, “Detector configuration of KAGRA, the Japanese cryogenic gravitational-wave detector,” Class. Quantum Grav. 29, 124007 (2012). [CrossRef]
3. G. D. Cole, W. Zhang, M. J. Martin, J. Ye, and M. Aspelmeyer, “Tenfold reduction of Brownian noise in high-reflectivity optical coatings,” Nature Photon. 7, 644 (2013). [CrossRef]
4. W. Gu, F. Wu, Y. Liu, and X. Hou, “Removal of surface figure deformation due to gravity in optical test,” Proc. SPIE 9282, 928209 (2014) [CrossRef]
5. E. Hirose, D. Bajuk, G. Billingsley, T. Kajita, B. Kestner, N. Mio, M. Ohashi, B. Reichman, H. Yamamoto, and L. Zhang, “Sapphire mirror for the KAGRA gravitational wave detector,” Phys. Rev. D 89, 062003 (2014). [CrossRef]
6. F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca, J. Amarni, P. Astone, G. Balestri, G. Ballardin, F. Barone, J.-P. Baronick, M. Barsuglia, A. Basti, F. Basti, T. S. Bauer, V. Bavigadda, M. Bejger, M. G. Beker, C. Belczynski, D. Bersanetti, A. Bertolini, M. Bitossi, M. A. Bizouard, S. Bloemen, M. Blom, M. Boer, G. Bogaert, D. Bondi, F. Bondu, L. Bonelli, R. Bonnand, V. Boschi, L. Bosi, T. Bouedo, C. Bradaschia, M. Branchesi, T. Briant, A. Brillet, V. Brisson, T. Bulik, H. J. Bulten, D. Buskulic, C. Buy, G. Cagnoli, E. Calloni, C. Campeggi, B. Canuel, F. Carbognani, F. Cavalier, R. Cavalieri, G. Cella, E. Cesarini, E. C. Mottin, A. Chincarini, A. Chiummo, S. Chua, F. Cleva, E. Coccia, P.-F. Cohadon, A. Colla, M. Colombini, A. Conte, J.-P. Coulon, E. Cuoco, A. Dalmaz, S. DAntonio, V. Dattilo, M. Davier, R. Day, G. Debreczeni, J. Degallaix, S. Delglise, W. D. Pozzo, H. Dereli, R. D. Rosa, L. D. Fiore, A. D. Lieto, A. D. Virgilio, M. Doets, V. Dolique, M. Drago, M. Ducrot, G. Endrczi, V. Fafone, S. Farinon, I. Ferrante, F. Ferrini, F. Fidecaro, I. Fiori, R. Flaminio, J.-D. Fournier, S. Franco, S. Frasca, F. Frasconi, L. Gammaitoni, F. Garufi, M. Gaspard, A. Gatto, G. Gemme, B. Gendre, E. Genin, A. Gennai, S. Ghosh, L. Giacobone, A. Giazotto, R. Gouaty, M. Granata, G. Greco, P. Groot, G. M. Guidi, J. Harms, A. Heidmann, H. Heitmann, P. Hello, G. Hemming, E. Hennes, D. Hofman, P. Jaranowski, R. J. G. Jonker, M. Kasprzack, F. Kflian, I. Kowalska, M. Kraan, A. Krlak, A. Kutynia, C. Lazzaro, M. Leonardi, N. Leroy, N. Letendre, T. G. F. Li, B. Lieunard, M. Lorenzini, V. Loriette, G. Losurdo, C. Magazz, E. Majorana, I. Maksimovic, V. Malvezzi, N. Man, V. Mangano, M. Mantovani, F. Marchesoni, F. Marion, J. Marque, F. Martelli, L. Martellini, A. Masserot, D. Meacher, J. Meidam, F. Mezzani, C. Michel, L. Milano, Y. Minenkov, A. Moggi, M. Mohan, M. Montani, N. Morgado, B. Mours, F. Mul, M. F. Nagy, I. Nardecchia, L. Naticchioni, G. Nelemans, I. Neri, M. Neri, F. Nocera, E. Pacaud, C. Palomba, F. Paoletti, A. Paoli, A. Pasqualetti, R. Passaquieti, D. Passuello, M. Perciballi, S. Petit, M. Pichot, F. Piergiovanni, G. Pillant, A. Piluso, L. Pinard, R. Poggiani, M. Prijatelj, G. A. Prodi, M. Punturo, P. Puppo, D. S. Rabeling, I. Rcz, P. Rapagnani, M. Razzano, V. Re, T. Regimbau, F. Ricci, F. Robinet, A. Rocchi, L. Rolland, R. Romano, D. Rosiska, P. Ruggi, E. Saracco, B. Sassolas, F. Schimmel, D. Sentenac, V. Sequino, S. Shah, K. Siellez, N. Straniero, B. Swinkels, M. Tacca, M. Tonelli, F. Travasso, M. Turconi, G. Vajente, N. v. Bakel, M. v. Beuzekom, J. F. J. v. d. Brand, C. V. D. Broeck, M. V. v. d. Sluys, J. v. Heijningen, M. Vasth, G. Vedovato, J. Veitch, D. Verkindt, F. Vetrano, A. Vicer, J.-Y. Vinet, G. Visser, H. Vocca, R. Ward, M. Was, L.-W. Wei, M. Yvert, A. Z. ny, and J.-P. Zendri, “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quantum Grav. 32, 024001 (2015). [CrossRef]
7. F. Siewert, J. Buchheim, T. Zeschke, G. Brenner, S. Kapitzki, and K. Tiedtke, “Sub-nm accuracy metrology for ultra-precise reflective X-ray optics,” Nucl. Instrum. Methods Phys. Res. A 635, S52 (2011). [CrossRef]
8. P. J. McGregor, G. J. Bloxham, R. Boz, J. Davies, M. Doolan, M. Ellis, J. Hart, D. J. Jones, L. Luvaul, J. Nielsen, S. Parcell, R. Sharp, D. Stevanovic, and P. J. Young, “GMT integral-field spectrograph (GMTIFS) conceptual design,” Proc. SPIE 8446, 84461I (2012). [CrossRef]
9. J. B. Oliver, J. Howe, A. Rigatti, D. J. Smith, and C. J. Stolz, “High precision coating technology for large aperture NIF optics,” in Optical Interference Coatings, OSA Technical Digest, p. ThD2-1 (2001).
10. A. V. Cumming, A. S. Bell, L. Barsotti, M. A. Barton, G. Cagnoli, D. Cook, L. Cunningham, M. Evans, G. D. Hammond, G. M. Harry, A. Heptonstall, J. Hough, R. Jones, R. Kumar, R. Mittleman, N. A. Robertson, S. Rowan, B. Shapiro, K. A. Strain, K. Tokmakov, C. Torrie, and A. A. van Veggel, ”Design and development of the advanced LIGO monolithic fused silica suspension,” Class. Quant. Grav. 29(3), 035003 (2012). [CrossRef]
11. COMSOL Multiphysics: http://www.comsol.com/comsol-multiphysics
12. J. Burke and U. Griesmann, “Calibrating the sag due to gravity of horizontal interferometer reference flat,” Proc. SPIE 8493, 84930F (2012) [CrossRef]
13. B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer”, Appl. Opt. 39, 5161 (2000) [CrossRef]
14. P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, and C. J. Walsh, “300-mm-aperture phase-shifting Fizeau interferometer,” Opt. Eng. 38, 1371 (1999). [CrossRef]
