Abstract

Thermal noise of optical reference cavities sets a fundamental limit to the frequency instability of ultrastable lasers. Using Levin’s formulation of the fluctuation-dissipation theorem, we correct the analytical estimate for the spacer contribution given by Numata et al. [Phys. Rev. Lett. 93, 250602 (2004)]. For detailed analysis, finite- element calculations of the thermal noise focusing on the spacer geometry, support structure, and the usage of different materials have been carried out. We find that the increased dissipation close to the contact area between spacer and mirrors can contribute significantly to the thermal noise. From an estimate of the support structure contribution, we give guidelines for a low-noise mounting of the cavity. For mixed-material cavities, we show that the thermal expansion can be compensated without deteriorating the thermal noise.

© 2011 Optical Society of America

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    [CrossRef] [PubMed]
  3. A. Bartels, S. A. Diddams, C. W. Oates, G. Wilpers, J. C. Bergquist, W. H. Oskay, and L. Hollberg, “Femtosecond-laser-based synthesis of ultrastable microwave signals from optical frequency references,” Opt. Lett. 30, 667–669 (2005).
    [CrossRef] [PubMed]
  4. J. Millo, M. Abgrall, M. Lours, E. English, H. Jiang, J. Guéna, A. Clairon, S. Bize, Y. L. Coq, G. Santarelli, and M. Tobar, “Ultra-low noise microwave generation with fiber-based optical frequency comb and application to atomic fountain clock,” Opt. Lett. 34, 3707–3709 (2009).
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  5. B. Lipphardt, G. Grosche, U. Sterr, C. Tamm, S. Weyers, and H. Schnatz, “The stability of an optical clock laser transferred to the interrogation oscillator for a Cs fountain,” IEEE Trans. Instrum. Meas. 58, 1258–1262 (2009).
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  7. H. Jiang, F. Kéfélian, S. Crane, O. Lopez, M. Lours, J. Millo, D. Holleville, P. Lemonde, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Transfer of an optical frequency over an urban fiber link,” J. Opt. Soc. Am. B 25, 2029–2035 (2008).
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  11. M. Notcutt, L.-S. Ma, A. D. Ludlow, S. M. Foreman, J. Ye, and J. L. Hall, “Contribution of thermal noise to frequency stability of rigid optical cavity via Hertz-linewidth lasers,” Phys. Rev. A 73, 031804 (2006).
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  38. F. Phelps, “Airy points of a meter bar,” Am. J. Phys. 34, 419–422(1966).
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  39. J. J. Wortman and R. A. Evans, “Young’s modulus, shear modulus, and Poisson’s ratio in silicon and germanium,” J. Appl. Phys. 36, 153–156 (1965).
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  40. D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys. 30, 621–629 (1978).
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    [CrossRef]
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  45. J. Giaime, P. Saha, D. Shoemaker, and L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
    [CrossRef]

2010 (3)

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104, 163903 (2010).
[CrossRef] [PubMed]

J. Lodewyck, P. G. Westergaard, A. Lecallier, L. Lorini, and P. Lemonde, “Frequency stability of optical lattice clocks,” New J. Phys. 12, 065026 (2010).
[CrossRef]

T. Legero, T. Kessler, and U. Sterr, “Tuning the thermal expansion properties of optical reference cavities with fused silica mirrors,” J. Opt. Soc. Am. B 27, 914–919 (2010).
[CrossRef]

2009 (5)

J. Millo, M. Abgrall, M. Lours, E. English, H. Jiang, J. Guéna, A. Clairon, S. Bize, Y. L. Coq, G. Santarelli, and M. Tobar, “Ultra-low noise microwave generation with fiber-based optical frequency comb and application to atomic fountain clock,” Opt. Lett. 34, 3707–3709 (2009).
[CrossRef] [PubMed]

J. Millo, D. V. Magalhães, C. Mandache, Y. Le Coq, E. M. L. English, P. G. Westergaard, J. Lodewyck, S. Bize, P. Lemonde, and G. Santarelli, “Ultrastable lasers based on vibration insensitive cavities,” Phys. Rev. A 79, 053829 (2009).
[CrossRef]

P. Dubé, A. Madej, J. Bernard, L. Marmet, and A. Shiner, “A narrow linewidth and frequency-stable probe laser source for the Sr+88 single ion optical frequency standard,” Appl. Phys. B 95, 43–54 (2009).
[CrossRef]

B. Lipphardt, G. Grosche, U. Sterr, C. Tamm, S. Weyers, and H. Schnatz, “The stability of an optical clock laser transferred to the interrogation oscillator for a Cs fountain,” IEEE Trans. Instrum. Meas. 58, 1258–1262 (2009).
[CrossRef]

O. Terra, G. Grosche, K. Predehl, R. Holzwarth, T. Legero, U. Sterr, B. Lipphardt, and H. Schnatz, “Phase-coherent comparison of two optical frequency standards over 146 km using a telecommunication fiber link,” Appl. Phys. B 97, 541–551 (2009).
[CrossRef]

2008 (9)

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319, 1805–1808 (2008).
[CrossRef] [PubMed]

G. D. Cole, S. Gröblacher, K. Gugler, S. Gigan, and M. Aspelmeyer, “Monocrystalline AlxGa1−xAs heterostructures for high-reflectivity high-Q micromechanical resonators in the megahertz regime,” Appl. Phys. Lett. 92, 261108 (2008).
[CrossRef]

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, “Thermal-noise-limited optical cavity,” Phys. Rev. A 77, 033847(2008).
[CrossRef]

M. Evans, S. Ballmer, M. Fejer, P. Fritschel, G. Harry, and G. Ogin, “Thermo-optic noise in coated mirrors for high-precision optical measurements,” Phys. Rev. D 78, 102003(2008).
[CrossRef]

Y. Levin, “Fluctuation-dissipation theorem for thermo-refractive noise,” Phys. Lett. A 372, 1941–1944 (2008).
[CrossRef]

M. L. Gorodetsky, “Thermal noises and noise compensation in high-reflection multilayer coating,” Phys. Lett. A 372, 6813–6822 (2008).
[CrossRef]

P. A. Williams, W. C. Swann, and N. R. Newbury, “High-stability transfer of an optical frequency over long fiber-optic links,” J. Opt. Soc. Am. B 25, 1284–1293 (2008).
[CrossRef]

H. Jiang, F. Kéfélian, S. Crane, O. Lopez, M. Lours, J. Millo, D. Holleville, P. Lemonde, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Transfer of an optical frequency over an urban fiber link,” J. Opt. Soc. Am. B 25, 2029–2035 (2008).
[CrossRef]

2007 (2)

2006 (4)

T. Nazarova, F. Riehle, and U. Sterr, “Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser,” Appl. Phys. B 83, 531–536 (2006).
[CrossRef]

M. Notcutt, L.-S. Ma, A. D. Ludlow, S. M. Foreman, J. Ye, and J. L. Hall, “Contribution of thermal noise to frequency stability of rigid optical cavity via Hertz-linewidth lasers,” Phys. Rev. A 73, 031804 (2006).
[CrossRef]

H. Stoehr, F. Mensing, J. Helmcke, and U. Sterr, “Diode laser with 1 Hz linewidth,” Opt. Lett. 31, 736–738 (2006).
[CrossRef] [PubMed]

B. Mours, E. Tournefier, and J.-Y. Vinet, “Thermal noise reduction in interferometric gravitational wave antennas: using high order TEM modes,” Class. Quantum Grav. 23, 5777–5784 (2006).
[CrossRef]

2005 (1)

2004 (2)

K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. 93, 250602 (2004).
[CrossRef]

M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, “Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors,” Phys. Rev. D 70, 082003 (2004).
[CrossRef]

2003 (2)

V. B. Braginsky and S. P. Vyatchanin, “Thermodynamical fluctuations in optical mirror coatings,” Phys. Lett. A 312, 244–255 (2003).
[CrossRef]

D. Hoffman, “Dynamic mechanical signatures of Viton A and plastic bonded explosives based on this polymer,” Polym. Eng. Sci. 43, 139–156 (2003).
[CrossRef]

2002 (1)

G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002).
[CrossRef]

2000 (1)

Y. T. Liu and K. S. Thorne, “Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses,” Phys. Rev. D 62, 122002 (2000).
[CrossRef]

1999 (2)

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, “Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae,” Phys. Lett. A 264, 1–10(1999).
[CrossRef]

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802(1999).
[CrossRef]

1998 (2)

Y. Levin, “Internal thermal noise in the LIGO test masses: a direct approach,” Phys. Rev. D 57, 659–663 (1998).
[CrossRef]

F. Bondu, P. Hello, and J.-Y. Vinet, “Thermal noise in mirrors of interferometric gravitational wave antennas,” Phys. Lett. A 246, 227–236 (1998).
[CrossRef]

1996 (1)

J. Giaime, P. Saha, D. Shoemaker, and L. Sievers, “A passive vibration isolation stack for LIGO: design, modeling, and testing,” Rev. Sci. Instrum. 67, 208–214 (1996).
[CrossRef]

1995 (1)

M. Notcutt, C. T. Taylor, A. G. Mann, and D. G. Blair, “Temperature compensation for cryogenic cavity stabilized lasers,” J. Phys. D 28, 1807–1810 (1995).
[CrossRef]

1978 (1)

D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys. 30, 621–629 (1978).
[CrossRef]

1966 (2)

F. Phelps, “Airy points of a meter bar,” Am. J. Phys. 34, 419–422(1966).
[CrossRef]

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

1965 (1)

J. J. Wortman and R. A. Evans, “Young’s modulus, shear modulus, and Poisson’s ratio in silicon and germanium,” J. Appl. Phys. 36, 153–156 (1965).
[CrossRef]

1951 (1)

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

Abgrall, M.

Allan, D. W.

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221–230 (1966).
[CrossRef]

Amy-Klein, A.

Aspelmeyer, M.

G. D. Cole, S. Gröblacher, K. Gugler, S. Gigan, and M. Aspelmeyer, “Monocrystalline AlxGa1−xAs heterostructures for high-reflectivity high-Q micromechanical resonators in the megahertz regime,” Appl. Phys. Lett. 92, 261108 (2008).
[CrossRef]

Ballmer, S.

M. Evans, S. Ballmer, M. Fejer, P. Fritschel, G. Harry, and G. Ogin, “Thermo-optic noise in coated mirrors for high-precision optical measurements,” Phys. Rev. D 78, 102003(2008).
[CrossRef]

Barber, Z. W.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319, 1805–1808 (2008).
[CrossRef] [PubMed]

Bartels, A.

Beck, K. M.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319, 1805–1808 (2008).
[CrossRef] [PubMed]

Bergquist, J. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[CrossRef] [PubMed]

A. Bartels, S. A. Diddams, C. W. Oates, G. Wilpers, J. C. Bergquist, W. H. Oskay, and L. Hollberg, “Femtosecond-laser-based synthesis of ultrastable microwave signals from optical frequency references,” Opt. Lett. 30, 667–669 (2005).
[CrossRef] [PubMed]

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802(1999).
[CrossRef]

Bernard, J.

P. Dubé, A. Madej, J. Bernard, L. Marmet, and A. Shiner, “A narrow linewidth and frequency-stable probe laser source for the Sr+88 single ion optical frequency standard,” Appl. Phys. B 95, 43–54 (2009).
[CrossRef]

Bize, S.

J. Millo, D. V. Magalhães, C. Mandache, Y. Le Coq, E. M. L. English, P. G. Westergaard, J. Lodewyck, S. Bize, P. Lemonde, and G. Santarelli, “Ultrastable lasers based on vibration insensitive cavities,” Phys. Rev. A 79, 053829 (2009).
[CrossRef]

J. Millo, M. Abgrall, M. Lours, E. English, H. Jiang, J. Guéna, A. Clairon, S. Bize, Y. L. Coq, G. Santarelli, and M. Tobar, “Ultra-low noise microwave generation with fiber-based optical frequency comb and application to atomic fountain clock,” Opt. Lett. 34, 3707–3709 (2009).
[CrossRef] [PubMed]

Blair, D. G.

M. Notcutt, C. T. Taylor, A. G. Mann, and D. G. Blair, “Temperature compensation for cryogenic cavity stabilized lasers,” J. Phys. D 28, 1807–1810 (1995).
[CrossRef]

Blatt, S.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319, 1805–1808 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, “Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10−15,” Opt. Lett. 32, 641–643 (2007).
[CrossRef] [PubMed]

Bondu, F.

F. Bondu, P. Hello, and J.-Y. Vinet, “Thermal noise in mirrors of interferometric gravitational wave antennas,” Phys. Lett. A 246, 227–236 (1998).
[CrossRef]

Boyd, M. M.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319, 1805–1808 (2008).
[CrossRef] [PubMed]

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A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319, 1805–1808 (2008).
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Clausnitzer, T.

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G. D. Cole, S. Gröblacher, K. Gugler, S. Gigan, and M. Aspelmeyer, “Monocrystalline AlxGa1−xAs heterostructures for high-reflectivity high-Q micromechanical resonators in the megahertz regime,” Appl. Phys. Lett. 92, 261108 (2008).
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M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, “Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors,” Phys. Rev. D 70, 082003 (2004).
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G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002).
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A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at 1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319, 1805–1808 (2008).
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T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
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M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, “Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors,” Phys. Rev. D 70, 082003 (2004).
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M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, “Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors,” Phys. Rev. D 70, 082003 (2004).
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G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002).
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D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys. 30, 621–629 (1978).
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M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, “Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors,” Phys. Rev. D 70, 082003 (2004).
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G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002).
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O. Terra, G. Grosche, K. Predehl, R. Holzwarth, T. Legero, U. Sterr, B. Lipphardt, and H. Schnatz, “Phase-coherent comparison of two optical frequency standards over 146 km using a telecommunication fiber link,” Appl. Phys. B 97, 541–551 (2009).
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M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, “Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors,” Phys. Rev. D 70, 082003 (2004).
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Hume, D. B.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
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T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
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F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104, 163903 (2010).
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D. F. McGuigan, C. C. Lam, R. Q. Gram, A. W. Hoffman, D. H. Douglass, and H. W. Gutche, “Measurements of the mechanical Q of single-crystal silicon at low temperatures,” J. Low Temp. Phys. 30, 621–629 (1978).
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J. Millo, D. V. Magalhães, C. Mandache, Y. Le Coq, E. M. L. English, P. G. Westergaard, J. Lodewyck, S. Bize, P. Lemonde, and G. Santarelli, “Ultrastable lasers based on vibration insensitive cavities,” Phys. Rev. A 79, 053829 (2009).
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P. Dubé, A. Madej, J. Bernard, L. Marmet, and A. Shiner, “A narrow linewidth and frequency-stable probe laser source for the Sr+88 single ion optical frequency standard,” Appl. Phys. B 95, 43–54 (2009).
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M. M. Fejer, S. Rowan, G. Cagnoli, D. R. M. Crooks, A. Gretarsson, G. M. Harry, J. Hough, S. D. Penn, P. H. Sneddon, and S. P. Vyatchanin, “Thermoelastic dissipation in inhomogeneous media: loss measurements and displacement noise in coated test masses for interferometric gravitational wave detectors,” Phys. Rev. D 70, 082003 (2004).
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T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
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T. Legero, T. Kessler, and U. Sterr, “Tuning the thermal expansion properties of optical reference cavities with fused silica mirrors,” J. Opt. Soc. Am. B 27, 914–919 (2010).
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B. Lipphardt, G. Grosche, U. Sterr, C. Tamm, S. Weyers, and H. Schnatz, “The stability of an optical clock laser transferred to the interrogation oscillator for a Cs fountain,” IEEE Trans. Instrum. Meas. 58, 1258–1262 (2009).
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B. Lipphardt, G. Grosche, U. Sterr, C. Tamm, S. Weyers, and H. Schnatz, “The stability of an optical clock laser transferred to the interrogation oscillator for a Cs fountain,” IEEE Trans. Instrum. Meas. 58, 1258–1262 (2009).
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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Sketch of the cavity model with dimensions used for the FEM simulations in this publication.

Fig. 2
Fig. 2

Deformation and contour plot of the elastic strain energy density in the mirror substrate and spacer for a Gaussian pressure profile with a 2 mm waist on the mirror surface. The color coding corresponds to the logarithm of the energy density ρ U in SI units.

Fig. 3
Fig. 3

Simulated strain energy in slices of 1 mm thickness compared to a homogeneously loaded spacer ( U 0 ) [see Eq. (8)]. The estimate by Numata et al. [18] (dashed–dotted line) and the analytic estimate according to Eq. (8) (solid line) are shown as references.

Fig. 4
Fig. 4

Strain energy as a function of spacer length for a spacer diameter of 32 mm . The estimate by Numata et al. [18] (dashed–dotted curve) as well as the analytic estimate according to Eq. (8) (solid curve) are shown as a reference.

Fig. 5
Fig. 5

Elastic strain energy as a function of spacer diameter for a spacer length of 100 mm . The estimate by Numata et al. [18] (dashed–dotted curve) as well as the analytic estimate according to Eq. (8) (solid curve) are shown as references.

Fig. 6
Fig. 6

Sketch of a cavity supported by four elastic support pads.

Fig. 7
Fig. 7

Contributions to the thermal noise of the optical length S x ( f ) of a cavity with and without a thermal expansion compensation ring.

Fig. 8
Fig. 8

Contributions to the thermal noise of the optical length S x ( f ) as function of the spacer length L.

Tables (3)

Tables Icon

Table 1 Parameters Used for the FEM Simulation

Tables Icon

Table 2 Comparison of Strain Energies in Nanojoules for an All-ULE Cavity

Tables Icon

Table 3 Parameters for Simulation of the Cavity Support

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

S x ( f ) = 2 k B T π 2 f 2 W diss F 0 2 ,
W diss = 2 π f U ϕ ,
S x ( f ) = 4 k B T π f F 0 2 ( U sp ϕ sp + 2 U sb ϕ sb + 2 U ct ϕ ct ) = S x ( sp ) ( f ) + 2 · S x ( sb ) ( f ) + 2 · S x ( ct ) ( f ) .
σ y = 2 ln ( 2 ) S y ( f ) f .
p ( r ) = ± 2 F 0 π w 2 e 2 r 2 / w 2 ,
U sb = 1 σ 2 2 π E w F 0 2 , S x ( sb ) ( f ) = 4 k B T π f 1 σ 2 2 π E w ϕ sb ,
U ct = U sb 2 π 1 2 σ 1 σ d ct w , S x ( ct ) ( f ) = S x ( sb ) ( f ) 2 π 1 2 σ 1 σ ϕ ct ϕ sb d ct w .
U sp ( 0 ) = L 2 E A sp F 0 2 = L 2 π E ( R sp 2 r sp 2 ) F 0 2 .
S L ( f ) = 4 k B T π f L 2 π E ( R sp 2 r sp 2 ) ϕ sp .
δ L sup = L sup A sp E sp F 0 .
U sup = 1 2 A sup G sup d sup δ L sup 2 A sup A sp L sup d sup G sup E sp L sup L U sp ( 0 ) .
S x ( sup ) ( f ) = 4 4 k B T π f F 0 2 U sup ϕ sup = 4 A sup A sp L sup d sup G sup E sp L sup L ϕ sup ϕ sp S L ( f ) .

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