Abstract

Scattered light in inteferometric gravitational wave detectors needs to be reduced so that it will not harm the actual signals coming from a gravitational wave. In this paper, we report on the application of the theory of light scattering from mirrors in interferometric detectors having multilayer coatings on their surfaces and compared the results with single-surface scattering theories, which are traditionally used in the field of gravitational wave detectors. For the first time in this field, we have calculated the scattering distributions of the power-recycling, the signal-recycling, and the beam-splitter mirrors in KAGRA (a cryogenic interferometric gravitational wave detector currently under construction in the Kamioka mine in Japan) by using models of their multilayer coatings. Furthermore, we have performed simulations to show the differences between multilayer scattering and single-surface scattering models in the back-scattering of mechanical structures close to the mirrors and the impact on the sensitivity of the KAGRA detector. We show that the back-scattering by using those coatings can be larger by up to almost two orders of magnitude and they also give rise to additional scattering features that should be taken into account for all optical applications in gravitational wave detectors.

© 2017 Optical Society of America

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References

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LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

M. Zerrad, S. Liukaityte, M. Lequime, and C. Amra, “Light scattered by optical coatings: numerical predictions and comparison to experiment for a global analysis,” Appl. Opt. 55, 9680–9687 (2016).
[Crossref] [PubMed]

2015 (4)

K. Kuroda, W.-T. Ni, and W.-P. Pan, “Gravitational waves: Classification, methods of detection, sensitivities and sources,” Int. J. Mod. Phys. D 24, 1530031 (2015).
[Crossref]

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

K. Kuroda, “Ground-based gravitational-wave detectors,” Int. J. Mod. Phys. D 24, 1530032 (2015).
[Crossref]

V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015).
[Crossref]

2014 (1)

2013 (2)

N. Choi and J. E. Harvey, “Numerical validation of the generalized harvey-shack surface scatter theory,” Opt. Eng. 52, 115103 (2013).
[Crossref]

Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
[Crossref]

2012 (2)

J. E. Harvey, S. Schröder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
[Crossref]

D. J. Ottaway, P. Fritschel, and S. J. Waldman, “Impact of upconverted scattered light on advanced interferometric gravitational wave detectors,” Opt. Express 20, 8329–8336 (2012).
[Crossref] [PubMed]

2011 (2)

1997 (1)

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[Crossref]

1996 (1)

S. Maure, G. Albrand, and C. Amra, “Low-level scattering and localized defects,” Appl. Opt. 35, 5572–5582 (1996).
[Crossref]

1994 (1)

1993 (2)

1992 (1)

1991 (1)

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71–86 (1991).
[Crossref]

1986 (1)

1981 (1)

P. Bousquet, F. Flory, and P. Rouche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. A 71, 1115–1124 (1981).
[Crossref]

1980 (1)

1978 (1)

A. R. Henderson, “The design of non-polarizing beam splitters,” Thin Solid Films 51, 339–347 (1978).
[Crossref]

Agatsuma, K.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Akutsu, T.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Albrand, G.

Amra, C.

Ando, M.

Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
[Crossref]

Apfel, J.

Arai, K.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Aso, Y.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
[Crossref]

Y. Aso, Y. Michimura, and K. Somiya, “Kagra main interferometer design document,” KAGRA internal document JGW-T1200913-v6 (2014).

Baccigalupi, C.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Barton, M.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Bennett, J.

Bertolini, A.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Bian, X.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Blair, D.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Bondu, F.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[Crossref]

Bousquet, P.

P. Bousquet, F. Flory, and P. Rouche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. A 71, 1115–1124 (1981).
[Crossref]

Braccini, S.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[Crossref]

Brisson, V.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[Crossref]

Bruel, L.

Cai, R.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Cao, Z.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Chang, Z.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Chao, S.

V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015).
[Crossref]

Choi, N.

N. Choi and J. E. Harvey, “Numerical validation of the generalized harvey-shack surface scatter theory,” Opt. Eng. 52, 115103 (2013).
[Crossref]

J. E. Harvey, S. Schröder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
[Crossref]

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surface with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A,  28, 1121–1139 (2011).
[Crossref]

Chu, Q.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Church, E. L.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71–86 (1991).
[Crossref]

E. L. Church and P. Z. Takacs, “Surface scattering,” in Handbook of Optics: Fundamentals, Techniques, and Design, M. Bass, ed.(McGraw-Hill, 1994), Vol. 1.

Cole, G.

V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015).
[Crossref]

Coriand, L.

Degallaix, J.

V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015).
[Crossref]

DeSalvo, R.

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F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
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F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
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Pan, W.-P.

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Pena-Arellano, F. E.

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Pinard, L.

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Reitze, D. H.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
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F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
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Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
[Crossref]

Shannon, R. N.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Shoda, A.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Sinzinger, S.

Somiya, K.

Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
[Crossref]

Y. Aso, Y. Michimura, and K. Somiya, “Kagra main interferometer design document,” KAGRA internal document JGW-T1200913-v6 (2014).

Sones, B. A.

B. A. Sones, “The optimization and analytical characterization of super cavity mirrors for use in the single atom laser experiment,” Ph.D. thesis (1997).

Takacs, P. Z.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71–86 (1991).
[Crossref]

E. L. Church and P. Z. Takacs, “Surface scattering,” in Handbook of Optics: Fundamentals, Techniques, and Design, M. Bass, ed.(McGraw-Hill, 1994), Vol. 1.

Takahashi, R.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Takamori, A.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Tang, W.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Tatsumi, D.

Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
[Crossref]

Thorne, K. S.

E. E. Flanagan and K. S. Thorne, “Noise due to light scattering in interferometric gravitational wave detectors. i: Handbook of formulae, and their derivations,” LIGO internal document (2011).

Tournie, E.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[Crossref]

Tuennermann, A.

Uchiyama, T.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

van den Brand, J.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

van Heijningen, J.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Vinet, J.-Y.

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
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von Finck, A.

Waldman, S. J.

Wan, X.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Wang, Y.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

wei Pan, H.

V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015).
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Wen, L.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Willke, B.

V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015).
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Wu, L.-A.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Xu, P.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

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D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

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Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
[Crossref]

H. Yamamoto, “Transfer functions of scattered light in advligo coc,” LIGO internal document T060073-x0 (2006).

Yamamoto, K.

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Zang, Y.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Zerrad, M.

Zhang, F.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Zhang, H.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Zhao, C.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Zhu, X.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
[Crossref]

Zhu, Z.

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
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Appl. Opt. (8)

Int. J. Mod. Phys. D (2)

K. Kuroda, W.-T. Ni, and W.-P. Pan, “Gravitational waves: Classification, methods of detection, sensitivities and sources,” Int. J. Mod. Phys. D 24, 1530031 (2015).
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J. Opt. Soc. Am. A (3)

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J. E. Harvey, S. Schröder, N. Choi, and A. Duparre, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2012).
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Opt. Express (2)

Phys. Rev. D (2)

J.-Y. Vinet, V. Brisson, S. Braccini, I. Ferrante, L. Pinard, F. Bondu, and E. Tournie, “Scattered light noise in gravitational wave interferometric detectors: A statistical approach,” Phys. Rev. D 56, 6085–6095 (1997).
[Crossref]

Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the kagra gravitational wave detector,” Phys. Rev. D 88, 043007 (2013).
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LIGO Scientific Collaboration and Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
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Proc. SPIE (1)

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71–86 (1991).
[Crossref]

Rev. Sci. Instrum. (1)

F. E. Pena-Arellano, T. Sekiguchi, Y. Fujii, R. Takahashi, M. Barton, N. Hirata, A. Shoda, J. van Heijningen, R. Flaminio, R. DeSalvo, K. Okutumi, T. Akutsu, Y. Aso, H. Ishizaki, N. Ohishi, K. Yamamoto, T. Uchiyama, O. Miyakawa, M. Kamiizumi, A. Takamori, E. Majorana, K. Agatsuma, E. Hennes, J. van den Brand, and A. Bertolini, “Characterization of the room temperature payload prototype for the cryogenic interferometric gravitational wave detector KAGRA,” Rev. Sci. Instrum. 87, 034501 (2016).
[Crossref] [PubMed]

Sci. China Phys. Mech. Astron. (2)

V. Mitrofanov, S. Chao, H. wei Pan, L.-C. Kuo, G. Cole, J. Degallaix, and B. Willke, “Technology for the next gravitational wave detectors,” Sci. China Phys. Mech. Astron. 58, 120404 (2015).
[Crossref]

D. Blair, L. Ju, C. Zhao, L. Wen, Q. Chu, Q. Fang, R. Cai, J. Gao, X. Lin, D. Liu, L.-A. Wu, Z. Zhu, D. H. Reitze, K. Arai, F. Zhang, R. Flaminio, X. Zhu, G. Hobbs, R. N. Manchester, R. N. Shannon, C. Baccigalupi, W. Gao, P. Xu, X. Bian, Z. Cao, Z. Chang, P. Dong, X. Gong, S. Huang, P. Ju, Z. Luo, L. Qiang, W. Tang, X. Wan, Y. Wang, S. Xu, Y. Zang, H. Zhang, Y.-K. Lau, and W.-T. Ni, “Gravitational wave astronomy: the current status,” Sci. China Phys. Mech. Astron. 58, 120402 (2015).
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Y. Aso, Y. Michimura, and K. Somiya, “Kagra main interferometer design document,” KAGRA internal document JGW-T1200913-v6 (2014).

E. L. Church and P. Z. Takacs, “Surface scattering,” in Handbook of Optics: Fundamentals, Techniques, and Design, M. Bass, ed.(McGraw-Hill, 1994), Vol. 1.

A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. thesis (University of Central Florida2000).

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Figures (9)

Fig. 1
Fig. 1 Cross-sectional and front views of the PR/SR-mirrors (left) and the BS-mirror (right) with their respective recoil masses. The diameter of the respective mirror is given in mm in the front view.
Fig. 2
Fig. 2 Comparison of the one-dimensional PSDs of the mirrors which were investigated in this paper and two measurements done for VIRGO mirrors at the LMA. Additionally, the fit done for the BS is drawn.
Fig. 3
Fig. 3 Spectral density of the seismic phase noise of the Kamioka mine (black) and the respective noise of the PR, SR, and BS mirrors, resulting from multiplying the TF with the seismic noise (see text). The phase noise horizontal to the ground is drawn in the left figure; the vertical one is drawn in the right figure. For comparability and illustration, the up-conversion (see text) of the respective phase-noise spectra is also shown.
Fig. 4
Fig. 4 Calculated BRDF · cos θsc (scattering probability density) as a function of the scattering latitude θ according to Eqs. (2) and (1) for the PR and SR mirrors by normal incidence. In case of the multilayers, we give also the profile perpendicular to the plane of incidence (POI).
Fig. 5
Fig. 5 The paths of light for the PR/SR mirrors (left) and the BS (right). While the (near) normal incidence for the PR/SR mirrors creates only single (rotational symmetric) scattering distributions, the AOI of 45° for the BS and its 50% transparency give rise to 8 different asymmetric scattering distributions, considering multilayer coatings.
Fig. 6
Fig. 6 Cross sectional view of the scattering probability-density distribution along the POI for the four cases of scattering on the HR surface of the BS mirror. The AOI is 45° for the two cases in the upper panel and 29.187° for the lower panel. Positive values for θ indicate scattering in the direction of the incoming light along the POI (ϕ = 0°).
Fig. 7
Fig. 7 Cross sectional view of the scattering probability distribution along the POI for the four cases of scattering on the AR surface of the BS mirror. The AOI is 45° for the two cases in the upper panel and 29.187° for the lower panel. Positive values for θ indicate scattering in the direction of the incoming light along the POI (ϕ = 0°).
Fig. 8
Fig. 8 Diagram of a scattering event with the principal geometrical quantities that are used in this paper. ϕin is set to be 0 without loss of generality.
Fig. 9
Fig. 9 Simplified diagram of the events that are discussed in this paper, with nomenclature used for the calculations. The blue arrows represent scattered light from the mirror while the red ones represent scattered light from the recoil mass. The red-dotted arrow is scattered light from the recoil mass that hits the laser-spot on the mirror and can couple back into the main beam.

Tables (4)

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Table 1 ABC parameters of the fits on the combined PSD1−D curves (see text for details).

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Table 2 List of the mirrors, their beam waist parameter wm, the solid angle toward it, the laser power incoming on each mirror, and the related G-factor. The reason for the doubled entries is that the mirrors face different solid angles ΔΩl for incident and reflected beams.

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Table 3 Values for d P sssc d Ω l in W/sr for the PR and SR mirrors calculated from the outcome of the simulations for the two models and recoil masses made of a Lambertian scatterer and titanium. The AOI is zero for all mirrors.

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Table 4 Values for d P sssc d Ω 1 in W/sr for the different scattering cases on the BS mirror calculated from the outcome of the simulations for the two models and recoil masses made of a Lambertian scatterer and titanium.

Equations (15)

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BRDF m ( θ in ; θ sc , ϕ sc ) = 16 π 2 λ 4 ( cos θ in + cos θ sc ) 2 Q PSD ( f x , f y ) ,
BRDF m ( θ in ; θ sc , ϕ sc ) = 4 π 2 λ 4 f pol i , j C i C j * γ i , j ( f x , f y )
f x = sin θ sc cos ϕ sc sin θ in λ f y = sin θ sc sin ϕ sc λ .
PSD 1 D = A [ 1 + ( B f ) 2 ] C / 2 f = f x 2 + f y 2 .
PSD 2 D = K A B [ 1 + ( B f ) 2 ] ( C + 1 ) / 2 K = 1 2 π Γ [ ( C + 1 ) / 2 ] Γ ( C / 2 ) .
BTDF m ( θ in ; θ sc , ϕ sc ) = 16 π 2 λ 4 ( n 0 cos θ in + n s cos θ sc ) 2 Q PSD ( f x , f y ) .
BTDF m ( θ in ; θ sc , ϕ sc ) = 4 π 2 λ 4 n s n 0 f pol i , j C i + C j + * γ i , j ( f x , f y ) .
f x = sin θ sc cos ϕ sc n 0 n s sin θ in λ f y = sin θ sc sin ϕ sc λ .
h rec ( f ) = | G | P sssc Φ ( f ) .
BRDF = L sc ( θ in , ϕ in ; θ sc , ϕ sc ) E in ( θ in , ϕ in ) .
L sc = 2 P sc A sc cos θ sc Ω sc , E i = P in A sc ,
d Ω sc = d A c cos θ sc r 2 d Ω c = d A sc cos θ sc r 2 ,
d L sssc = BSDF m d E ssc d L sssc = BSDF m L ssc cos θ sc d Ω sc .
L ssc = 2 P ssc A m cos θ sc Ω sc ,
d L sssc = BSDF m 2 P ssc A m Ω sc d Ω sc .

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