Abstract

Ultra-high sensitivity temperature sensing and stable thermal control are crucial for many science experiments testing fundamental theories to high precision. Here we report the first pico-kevin scale thermometer operating at room temperature with an exceptionally low theoretical noise figure of ~70pK/Hz at 1 Hz and a high dynamic range of ~500 K. We have experimentally demonstrated a temperature sensitivity of <3.8nK/Hz at 1 Hz near room temperature, which is an order of magnitude improvement over the state of the art. We have also demonstrated an ultra-high stability thermal control system using this thermometer, achieving 3.7 nK stability at 1 s and ∼ 120 pK at 104 s, which is 10–100 times more stable than the state of the art. With some upgrades to this proof-of-principle device, we can expect it to be used for very high resolution tests of special relativity and in critical point phenomena.

© 2017 Optical Society of America

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References

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2015 (1)

2014 (2)

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

D. Long, A. Fleisher, S. Wójtewicz, and J. Hodges, “Quantum-noise-limited cavity ring-down spectroscopy,” Appl. Phys. B 115, 149–153 (2014).
[Crossref]

2012 (1)

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

2011 (3)

2009 (1)

Y. Yagodzinskyy, E. Andronova, M. Ivanchenko, and H. Hänninen, “Anelastic mechanical loss spectrometry of hydrogen in austenitic stainless steels,” Mater. Sci. Eng. A 521, 159–162 (2009).
[Crossref]

2008 (1)

2007 (3)

Y. Liu and L. Wei, “Low-cost high-sensitivity strain and temperature sensing using graded-index multimode fibers,” Appl. Opt. 46, 2516–2519 (2007).
[Crossref] [PubMed]

S. Webster, M. Oxborrow, and P. Gill, “Vibration insensitive optical cavity,” Phys. Rev. A 75, 011801 (2007).
[Crossref]

M. Barmatz, I. Hahn, J. Lipa, and R. Duncan, “Critical phenomena in microgravity: Past, present, and future,” Rev. Mod. Phys. 79, 1 (2007).
[Crossref]

2005 (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

2004 (1)

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]

2003 (2)

J. Lipa, J. Nissen, D. Stricker, D. Swanson, and T. Chui, “Specific heat of liquid helium in zero gravity very near the lambda point,” Phys. Rev. B 68, 174518 (2003).
[Crossref]

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9, 57–79 (2003).
[Crossref]

2002 (1)

N. Nakagawa, A. Gretarsson, E. Gustafson, and M. Fejer, “Thermal noise in half-infinite mirrors with nonuniform loss: A slab of excess loss in a half-infinite mirror,” Phys. Rev. D 65, 102001 (2002).
[Crossref]

2001 (2)

D. Beysens and Y. Garrabos, “Near-critical fluids under microgravity: status of the eseme program and perspectives for the iss,” Acta Astronaut. 48, 629–638 (2001).
[Crossref]

E. D. Black, “An introduction to pound–drever–hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001).
[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 (1)

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

1998 (1)

Y. Levin, “Internal thermal noise in the ligo test masses: A direct approach,” Phys. Rev. D 57, 659 (1998).
[Crossref]

1995 (1)

K. Koo and A. Kersey, “Bragg grating-based laser sensors systems with interferometric interrogation and wavelength division multiplexing: Optical fiber sensors,” J. Lightwave Technol. 13, 1243–1249 (1995).
[Crossref]

1993 (1)

A. Araya, K. Kawabe, T. Sato, N. Mio, and K. Tsubono, “Highly sensitive wideband seismometer using a laser interferometer,” Rev. Sci. Instrum. 64, 1337–1341 (1993).
[Crossref]

1983 (1)

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

1979 (2)

M. Moldover, J. Sengers, R. Gammon, and R. Hocken, “Gravity effects in fluids near the gas-liquid critical point,” Rev. Mod. Phys. 51, 79 (1979).
[Crossref]

G. Hocker, “Fiber-optic sensing of pressure and temperature,” Appl. Opt. 18, 1445–1448 (1979).
[Crossref] [PubMed]

1978 (1)

S. Wood, B. Mangum, J. Filliben, and S. Tillet, “An investigation of the stability of thermistors,” J. Res. Natl. Bur. Stand. 83, 247–263 (1978).
[Crossref]

1974 (1)

L. D. Bowers and P. W. Carr, “Noise measurement and the temperature resolution of negative temperature coefficient thermistors,” Thermochim. Acta 10, 129–142 (1974).
[Crossref]

Akahane, Y.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Alfauwaz, A.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

Andronova, E.

Y. Yagodzinskyy, E. Andronova, M. Ivanchenko, and H. Hänninen, “Anelastic mechanical loss spectrometry of hydrogen in austenitic stainless steels,” Mater. Sci. Eng. A 521, 159–162 (2009).
[Crossref]

Anstie, J. D.

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

Araya, A.

A. Araya, K. Kawabe, T. Sato, N. Mio, and K. Tsubono, “Highly sensitive wideband seismometer using a laser interferometer,” Rev. Sci. Instrum. 64, 1337–1341 (1993).
[Crossref]

Asano, T.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Banovic, S. W.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

Barmatz, M.

M. Barmatz, I. Hahn, J. Lipa, and R. Duncan, “Critical phenomena in microgravity: Past, present, and future,” Rev. Mod. Phys. 79, 1 (2007).
[Crossref]

Bass, M.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Baumgartel, L.

Baynes, F. N.

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

Beysens, D.

D. Beysens and Y. Garrabos, “Near-critical fluids under microgravity: status of the eseme program and perspectives for the iss,” Acta Astronaut. 48, 629–638 (2001).
[Crossref]

Black, E. D.

E. D. Black, “An introduction to pound–drever–hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001).
[Crossref]

Boukari, H.

R. W. Gammon, J. Shaumeyer, M. E. Briggs, H. Boukari, D. A. Gent, and R. A. Wilkinson, “Highlights of the zeno results from the usmp-2 mission, NASA technical memorandum 107031,” (1995).

Bowers, L. D.

L. D. Bowers and P. W. Carr, “Noise measurement and the temperature resolution of negative temperature coefficient thermistors,” Thermochim. Acta 10, 129–142 (1974).
[Crossref]

Braginsky, V.

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

Briggs, M. E.

R. W. Gammon, J. Shaumeyer, M. E. Briggs, H. Boukari, D. A. Gent, and R. A. Wilkinson, “Highlights of the zeno results from the usmp-2 mission, NASA technical memorandum 107031,” (1995).

Buchman, S.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

Byer, R.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

Camp, J.

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]

Campbell, G.

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

Carr, P. W.

L. D. Bowers and P. W. Carr, “Noise measurement and the temperature resolution of negative temperature coefficient thermistors,” Thermochim. Acta 10, 129–142 (1974).
[Crossref]

Chang, H.-J.

H.-J. Chang, Y.-C. Zheng, C.-L. Ma, and C.-L. Lee, “Highly sensitive fiber-optic thermometer using an air micro-bubble in a liquid core fiber fabry-pérot interferometer,” in “OptoElectronics and Communications Conference and Photonics in Switching,” (Optical Society of America, 2013), p. TuPS_16.

Chen, L.

L. Chen, “High-precision spectroscopy of molecular iodine: From optical frequency standards to global descriptions of hyperfine interactions and associated electronic structure,” Ph.D. thesis, Department of Physics, University of Colorado (2005).

Chui, T.

J. Lipa, J. Nissen, D. Stricker, D. Swanson, and T. Chui, “Specific heat of liquid helium in zero gravity very near the lambda point,” Phys. Rev. B 68, 174518 (2003).
[Crossref]

Chung, Y.

Conklin, J.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

Cutler, G.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

Davis, J. R.

J. R. Davis, ASM Specialty Handbook: Heat-Resistant Materials (Asm International, 1997).

DeCusatis, C.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Dolesi, R.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Dong, X.

Drever, R.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Duncan, R.

M. Barmatz, I. Hahn, J. Lipa, and R. Duncan, “Critical phenomena in microgravity: Past, present, and future,” Rev. Mod. Phys. 79, 1 (2007).
[Crossref]

Enoch, J.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Evans, M.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Fejer, M.

N. Nakagawa, A. Gretarsson, E. Gustafson, and M. Fejer, “Thermal noise in half-infinite mirrors with nonuniform loss: A slab of excess loss in a half-infinite mirror,” Phys. Rev. D 65, 102001 (2002).
[Crossref]

Fields, R. J.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

Filliben, J.

S. Wood, B. Mangum, J. Filliben, and S. Tillet, “An investigation of the stability of thermistors,” J. Res. Natl. Bur. Stand. 83, 247–263 (1978).
[Crossref]

Fleisher, A.

D. Long, A. Fleisher, S. Wójtewicz, and J. Hodges, “Quantum-noise-limited cavity ring-down spectroscopy,” Appl. Phys. B 115, 149–153 (2014).
[Crossref]

Foecke, T.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

Ford, G.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Fritschel, P.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Frost, M. H.J.

M. H.J. Frost, Deformation-Mechanism Maps: The Plasticity and Creep of Metals and CeramicsPergamon PressOxford [Oxford-shire]1982).

Gammon, R.

M. Moldover, J. Sengers, R. Gammon, and R. Hocken, “Gravity effects in fluids near the gas-liquid critical point,” Rev. Mod. Phys. 51, 79 (1979).
[Crossref]

Gammon, R. W.

R. W. Gammon, J. Shaumeyer, M. E. Briggs, H. Boukari, D. A. Gent, and R. A. Wilkinson, “Highlights of the zeno results from the usmp-2 mission, NASA technical memorandum 107031,” (1995).

Garrabos, Y.

D. Beysens and Y. Garrabos, “Near-critical fluids under microgravity: status of the eseme program and perspectives for the iss,” Acta Astronaut. 48, 629–638 (2001).
[Crossref]

Gayle, F. W.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

Gent, D. A.

R. W. Gammon, J. Shaumeyer, M. E. Briggs, H. Boukari, D. A. Gent, and R. A. Wilkinson, “Highlights of the zeno results from the usmp-2 mission, NASA technical memorandum 107031,” (1995).

Gill, P.

S. Webster, M. Oxborrow, and P. Gill, “Vibration insensitive optical cavity,” Phys. Rev. A 75, 011801 (2007).
[Crossref]

Gorodetsky, M.

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

Gretarsson, A.

N. Nakagawa, A. Gretarsson, E. Gustafson, and M. Fejer, “Thermal noise in half-infinite mirrors with nonuniform loss: A slab of excess loss in a half-infinite mirror,” Phys. Rev. D 65, 102001 (2002).
[Crossref]

Grudinin, I.

Guo, J.

Gustafson, E.

N. Nakagawa, A. Gretarsson, E. Gustafson, and M. Fejer, “Thermal noise in half-infinite mirrors with nonuniform loss: A slab of excess loss in a half-infinite mirror,” Phys. Rev. D 65, 102001 (2002).
[Crossref]

Hahn, I.

M. Barmatz, I. Hahn, J. Lipa, and R. Duncan, “Critical phenomena in microgravity: Past, present, and future,” Rev. Mod. Phys. 79, 1 (2007).
[Crossref]

Hall, J. L.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Han, M.

Hänninen, H.

Y. Yagodzinskyy, E. Andronova, M. Ivanchenko, and H. Hänninen, “Anelastic mechanical loss spectrometry of hydrogen in austenitic stainless steels,” Mater. Sci. Eng. A 521, 159–162 (2009).
[Crossref]

He, S.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Hocken, R.

M. Moldover, J. Sengers, R. Gammon, and R. Hocken, “Gravity effects in fluids near the gas-liquid critical point,” Rev. Mod. Phys. 51, 79 (1979).
[Crossref]

Hocker, G.

Hodges, J.

D. Long, A. Fleisher, S. Wójtewicz, and J. Hodges, “Quantum-noise-limited cavity ring-down spectroscopy,” Appl. Phys. B 115, 149–153 (2014).
[Crossref]

Hopper, D. J.

D. J. Hopper, “Investigation of laser frequency stabilisation using modulation transfer spectroscopy,” Ph.D. thesis, School of Physical and Chemical Sciences, Queensland University of Technologies (2008).

Hou, W.

Hough, J.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Hueller, M.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Hwang, D.

Ivanchenko, M.

Y. Yagodzinskyy, E. Andronova, M. Ivanchenko, and H. Hänninen, “Anelastic mechanical loss spectrometry of hydrogen in austenitic stainless steels,” Mater. Sci. Eng. A 521, 159–162 (2009).
[Crossref]

Jin, S.

Kawabe, K.

A. Araya, K. Kawabe, T. Sato, N. Mio, and K. Tsubono, “Highly sensitive wideband seismometer using a laser interferometer,” Rev. Sci. Instrum. 64, 1337–1341 (1993).
[Crossref]

Kemery, A.

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]

Kersey, A.

K. Koo and A. Kersey, “Bragg grating-based laser sensors systems with interferometric interrogation and wavelength division multiplexing: Optical fiber sensors,” J. Lightwave Technol. 13, 1243–1249 (1995).
[Crossref]

Kleinert, H.

H. Kleinert and V. Schulte-Frohlinde, “Critical properties of ϕ 4-theories,” World Scientific, Singapore (2001).
[Crossref]

Koo, K.

K. Koo and A. Kersey, “Bragg grating-based laser sensors systems with interferometric interrogation and wavelength division multiplexing: Optical fiber sensors,” J. Lightwave Technol. 13, 1243–1249 (1995).
[Crossref]

Kowalski, F.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Lakshminarayanan, V.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Le, J.

E. Marquardt, J. Le, and R. Radebaugh, “Cryogenic Material Properties Database,” in “Cryocoolers 11,” (Springer, 2002), pp. 681–687.

Lee, B.

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9, 57–79 (2003).
[Crossref]

Lee, C.-L.

H.-J. Chang, Y.-C. Zheng, C.-L. Ma, and C.-L. Lee, “Highly sensitive fiber-optic thermometer using an air micro-bubble in a liquid core fiber fabry-pérot interferometer,” in “OptoElectronics and Communications Conference and Photonics in Switching,” (Optical Society of America, 2013), p. TuPS_16.

Levin, Y.

Y. Levin, “Internal thermal noise in the ligo test masses: A direct approach,” Phys. Rev. D 57, 659 (1998).
[Crossref]

Li, G.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Lipa, J.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

M. Barmatz, I. Hahn, J. Lipa, and R. Duncan, “Critical phenomena in microgravity: Past, present, and future,” Rev. Mod. Phys. 79, 1 (2007).
[Crossref]

J. Lipa, J. Nissen, D. Stricker, D. Swanson, and T. Chui, “Specific heat of liquid helium in zero gravity very near the lambda point,” Phys. Rev. B 68, 174518 (2003).
[Crossref]

Liu, G.

Liu, Y.

Liu, Y. T.

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]

Long, D.

D. Long, A. Fleisher, S. Wójtewicz, and J. Hodges, “Quantum-noise-limited cavity ring-down spectroscopy,” Appl. Phys. B 115, 149–153 (2014).
[Crossref]

Luecke, W. E.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

Luiten, A. N.

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

Ma, C.-L.

H.-J. Chang, Y.-C. Zheng, C.-L. Ma, and C.-L. Lee, “Highly sensitive fiber-optic thermometer using an air micro-bubble in a liquid core fiber fabry-pérot interferometer,” in “OptoElectronics and Communications Conference and Photonics in Switching,” (Optical Society of America, 2013), p. TuPS_16.

Macdonald, C.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Mahajan, V.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Mangum, B.

S. Wood, B. Mangum, J. Filliben, and S. Tillet, “An investigation of the stability of thermistors,” J. Res. Natl. Bur. Stand. 83, 247–263 (1978).
[Crossref]

Marquardt, E.

E. Marquardt, J. Le, and R. Radebaugh, “Cryogenic Material Properties Database,” in “Cryocoolers 11,” (Springer, 2002), pp. 681–687.

McColskey, J. D.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

McCowan, C. N.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

Mio, N.

A. Araya, K. Kawabe, T. Sato, N. Mio, and K. Tsubono, “Highly sensitive wideband seismometer using a laser interferometer,” Rev. Sci. Instrum. 64, 1337–1341 (1993).
[Crossref]

Moldover, M.

M. Moldover, J. Sengers, R. Gammon, and R. Hocken, “Gravity effects in fluids near the gas-liquid critical point,” Rev. Mod. Phys. 51, 79 (1979).
[Crossref]

Moon, D. S.

Moon, S.

Munley, A.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Nakagawa, N.

N. Nakagawa, A. Gretarsson, E. Gustafson, and M. Fejer, “Thermal noise in half-infinite mirrors with nonuniform loss: A slab of excess loss in a half-infinite mirror,” Phys. Rev. D 65, 102001 (2002).
[Crossref]

Nguyen, L. V.

Nicolodi, D.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Nissen, J.

J. Lipa, J. Nissen, D. Stricker, D. Swanson, and T. Chui, “Specific heat of liquid helium in zero gravity very near the lambda point,” Phys. Rev. B 68, 174518 (2003).
[Crossref]

Noda, S.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Numata, K.

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]

Oxborrow, M.

S. Webster, M. Oxborrow, and P. Gill, “Vibration insensitive optical cavity,” Phys. Rev. A 75, 011801 (2007).
[Crossref]

Qian, W.

Radebaugh, R.

E. Marquardt, J. Le, and R. Radebaugh, “Cryogenic Material Properties Database,” in “Cryocoolers 11,” (Springer, 2002), pp. 681–687.

Saraf, S.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

Sato, T.

A. Araya, K. Kawabe, T. Sato, N. Mio, and K. Tsubono, “Highly sensitive wideband seismometer using a laser interferometer,” Rev. Sci. Instrum. 64, 1337–1341 (1993).
[Crossref]

Schulte-Frohlinde, V.

H. Kleinert and V. Schulte-Frohlinde, “Critical properties of ϕ 4-theories,” World Scientific, Singapore (2001).
[Crossref]

Sengers, J.

M. Moldover, J. Sengers, R. Gammon, and R. Hocken, “Gravity effects in fluids near the gas-liquid critical point,” Rev. Mod. Phys. 51, 79 (1979).
[Crossref]

Shaumeyer, J.

R. W. Gammon, J. Shaumeyer, M. E. Briggs, H. Boukari, D. A. Gent, and R. A. Wilkinson, “Highlights of the zeno results from the usmp-2 mission, NASA technical memorandum 107031,” (1995).

Siewert, T. A.

W. E. Luecke, J. D. McColskey, C. N. McCowan, S. W. Banovic, R. J. Fields, T. Foecke, T. A. Siewert, and F. W. Gayle, “Mechanical properties of structural steels (draft),” (2005).

Song, B.-S.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Stace, T. M.

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

Strekalov, D.

Stricker, D.

J. Lipa, J. Nissen, D. Stricker, D. Swanson, and T. Chui, “Specific heat of liquid helium in zero gravity very near the lambda point,” Phys. Rev. B 68, 174518 (2003).
[Crossref]

Swanson, D.

J. Lipa, J. Nissen, D. Stricker, D. Swanson, and T. Chui, “Specific heat of liquid helium in zero gravity very near the lambda point,” Phys. Rev. B 68, 174518 (2003).
[Crossref]

Tan, S.

S. Tan, “Sub nano kelvin thermometry and temperature stabilization using resonant cavity thermometry,” Ph.D. thesis, Department of Mechanical Engineering, Stanford University, ( https://searchworks.stanford.edu/view/11849465 ) (2016).

Thomas, R. C.

R. C. Thomas, “Quantum noise and radiation pressure effects in high power optical interferometers,” Ph.D. thesis, Department of Physics, MIT.

Thompson, R.

Thorne, K. S.

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]

Tillet, S.

S. Wood, B. Mangum, J. Filliben, and S. Tillet, “An investigation of the stability of thermistors,” J. Res. Natl. Bur. Stand. 83, 247–263 (1978).
[Crossref]

Tombolato, D.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Tsubono, K.

A. Araya, K. Kawabe, T. Sato, N. Mio, and K. Tsubono, “Highly sensitive wideband seismometer using a laser interferometer,” Rev. Sci. Instrum. 64, 1337–1341 (1993).
[Crossref]

Van Stryland, E.

M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, Handbook of Optics, Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry (McGraw-Hill, Inc., 2009).

Vitale, S.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Vyatchanin, S.

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

Ward, H.

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

Wass, P.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Weber, W.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Webster, S.

S. Webster, M. Oxborrow, and P. Gill, “Vibration insensitive optical cavity,” Phys. Rev. A 75, 011801 (2007).
[Crossref]

Wei, H.

Wei, L.

Weiss, R.

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

Weng, W.

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

Whitcomb, S. E.

M. E. Zucker and S. E. Whitcomb, “Measurement of optical path fluctuations due to residual gas in the ligo 40 meter interferometer,” in “Proc. 7th Marcel Grossman Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories,” (1996), pp. 1434–1436.

Whitcomb, S.E.

S.E. Whitcomb, “Optical pathlength fluctuations in an interferometer due to residual gas,” Unpublished.

Wilkinson, R. A.

R. W. Gammon, J. Shaumeyer, M. E. Briggs, H. Boukari, D. A. Gent, and R. A. Wilkinson, “Highlights of the zeno results from the usmp-2 mission, NASA technical memorandum 107031,” (1995).

Wójtewicz, S.

D. Long, A. Fleisher, S. Wójtewicz, and J. Hodges, “Quantum-noise-limited cavity ring-down spectroscopy,” Appl. Phys. B 115, 149–153 (2014).
[Crossref]

Wood, S.

S. Wood, B. Mangum, J. Filliben, and S. Tillet, “An investigation of the stability of thermistors,” J. Res. Natl. Bur. Stand. 83, 247–263 (1978).
[Crossref]

Yagodzinskyy, Y.

Y. Yagodzinskyy, E. Andronova, M. Ivanchenko, and H. Hänninen, “Anelastic mechanical loss spectrometry of hydrogen in austenitic stainless steels,” Mater. Sci. Eng. A 521, 159–162 (2009).
[Crossref]

Yamamoto, K.

K. Yamamoto, “Study of the thermal noise caused by inhomogeneously distributed loss,” Ph.D. thesis, Department of Physics, Graduate school of Science, University of Tokyo (2000).

Yu, N.

Zhang, S.

Zhang, Z.

Zhao, C.-L.

Zheng, Y.-C.

H.-J. Chang, Y.-C. Zheng, C.-L. Ma, and C.-L. Lee, “Highly sensitive fiber-optic thermometer using an air micro-bubble in a liquid core fiber fabry-pérot interferometer,” in “OptoElectronics and Communications Conference and Photonics in Switching,” (Optical Society of America, 2013), p. TuPS_16.

Zhou, J.

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

Zucker, M. E.

M. E. Zucker and S. E. Whitcomb, “Measurement of optical path fluctuations due to residual gas in the ligo 40 meter interferometer,” in “Proc. 7th Marcel Grossman Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories,” (1996), pp. 1434–1436.

Acta Astronaut. (1)

D. Beysens and Y. Garrabos, “Near-critical fluids under microgravity: status of the eseme program and perspectives for the iss,” Acta Astronaut. 48, 629–638 (2001).
[Crossref]

Am. J. Phys. (1)

E. D. Black, “An introduction to pound–drever–hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (2)

D. Long, A. Fleisher, S. Wójtewicz, and J. Hodges, “Quantum-noise-limited cavity ring-down spectroscopy,” Appl. Phys. B 115, 149–153 (2014).
[Crossref]

R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

arXiv preprint (1)

J. Lipa, S. Buchman, S. Saraf, J. Zhou, A. Alfauwaz, J. Conklin, G. Cutler, and R. Byer, “Prospects for an advanced kennedy-thorndike experiment in low earth orbit,” arXiv preprint arXiv:12033914 (2012).

J. Lightwave Technol. (1)

K. Koo and A. Kersey, “Bragg grating-based laser sensors systems with interferometric interrogation and wavelength division multiplexing: Optical fiber sensors,” J. Lightwave Technol. 13, 1243–1249 (1995).
[Crossref]

J. Res. Natl. Bur. Stand. (1)

S. Wood, B. Mangum, J. Filliben, and S. Tillet, “An investigation of the stability of thermistors,” J. Res. Natl. Bur. Stand. 83, 247–263 (1978).
[Crossref]

Mater. Sci. Eng. A (1)

Y. Yagodzinskyy, E. Andronova, M. Ivanchenko, and H. Hänninen, “Anelastic mechanical loss spectrometry of hydrogen in austenitic stainless steels,” Mater. Sci. Eng. A 521, 159–162 (2009).
[Crossref]

Nat. Mater. (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[Crossref]

Opt. Express (3)

Opt. Fiber Technol. (1)

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9, 57–79 (2003).
[Crossref]

Opt. Lett. (1)

Phys. Lett. A (1)

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

Phys. Rev. A (1)

S. Webster, M. Oxborrow, and P. Gill, “Vibration insensitive optical cavity,” Phys. Rev. A 75, 011801 (2007).
[Crossref]

Phys. Rev. B (1)

J. Lipa, J. Nissen, D. Stricker, D. Swanson, and T. Chui, “Specific heat of liquid helium in zero gravity very near the lambda point,” Phys. Rev. B 68, 174518 (2003).
[Crossref]

Phys. Rev. D (4)

N. Nakagawa, A. Gretarsson, E. Gustafson, and M. Fejer, “Thermal noise in half-infinite mirrors with nonuniform loss: A slab of excess loss in a half-infinite mirror,” Phys. Rev. D 65, 102001 (2002).
[Crossref]

Y. Levin, “Internal thermal noise in the ligo test masses: A direct approach,” Phys. Rev. D 57, 659 (1998).
[Crossref]

R. Dolesi, M. Hueller, D. Nicolodi, D. Tombolato, S. Vitale, P. Wass, W. Weber, M. Evans, P. Fritschel, R. Weiss, and et al., “Brownian force noise from molecular collisions and the sensitivity of advanced gravitational wave observatories,” Phys. Rev. D 84, 063007 (2011).
[Crossref]

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]

Phys. Rev. Lett. (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]

W. Weng, J. D. Anstie, T. M. Stace, G. Campbell, F. N. Baynes, and A. N. Luiten, “Nano-kelvin thermometry and temperature control: beyond the thermal noise limit,” Phys. Rev. Lett. 112, 160801 (2014).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

M. Moldover, J. Sengers, R. Gammon, and R. Hocken, “Gravity effects in fluids near the gas-liquid critical point,” Rev. Mod. Phys. 51, 79 (1979).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1

Conceptual diagram of resonant cavity thermometer. Orange: temperature sensor; blue: reference system. Detected signal frequency at photodetector (PD) is | f1f2|.

Fig. 2
Fig. 2

(a): Cutaway CAD view of the thermal test system. (b): Schematic of experimental setup. Optical paths are shown as solid lines, electrical signals as dashed lines. EOM is electro-optic modulator, LPF is low pass filter, PI is proportional integral controller and λ/4 is quarter wave plate. The lasers are stabilized to the ULE cavity using the PDH locking technique, and to the steel cavity with the dither lock technique [23]. Inputs 1, 2 can be the frequency reference from cavities for normal operation, or from the Iodine setup for calibration of each cavity.

Fig. 3
Fig. 3

Measured and theoretical vertical acceleration spectrum. Insert: schematic showing the test arrangement with the accelerometer mounted inside the thermal enclosure.

Fig. 4
Fig. 4

Amplitude spectrum of noise sources. Left axis is expressed in frequency noise amplitude, right axis is the equivalent temperature noise amplitude.

Fig. 5
Fig. 5

RCT thermal control setup. Optical signals are indicated by solid lines, electrical signals by dashed lines. Blue dashed lines show the laser stabilization control loop and black dashed lines indicate the thermal control loop. Green circle is the thermistor on the cavity spacer.

Fig. 6
Fig. 6

Comparison of RCT and thermistor outputs for various sized temperature steps. (a): A 52-hour data set with various step sizes. (b): A 4-hour section of the data showing detailed features of RCT and thermistor readings.

Fig. 7
Fig. 7

Expected temperature sensitivity of the RCT in Hz/nK over a wide range. Note that the sensitivity drops to zero at 23.6 K as the CTE of SS reaches a null point. Below the null it increases again to 1 Hz/nK near 4 K. The blue curve is calculated from a cryogenics model of the CTE for the temperature range of 4K to 293 K [41], and the orange curve is from a model spanning 311 K to 922 K [42].

Fig. 8(a)
Fig. 8(a)

where the Fig. 8. (a): Amplitude spectral density of RCT, with and without thermal feedback control activated; (b): Allan deviations of RCT. Grey line indicates a slope of 1 τ where τ is the averaging time.

Fig. 9
Fig. 9

(a): Linear correlation measurement between thermistor and RCT. The cooling and heating processes are differentiated by blue circles and orange dots. TR and TRCT are the relative temperature readings of thermistor and RCT, respectively. The uncertainties in the insert equation are obtained from the 95% confidence intervals of the fit. (b): Residuals of data shown in (a) after subtracting the linear fit. Error bars are obtained as described in the appendix.

Fig. 10
Fig. 10

AD of Thermistor electronics.

Fig. 11
Fig. 11

Temperature distribution along one half of the RCT cavity assuming the heater is located at the center. The x axis is the normalized cavity length, with 0 being cavity center. Orange: 2D axisymmetrical FEA simulation, with symmetrical boundary condition at x/L = 0, and radiation rate equals conduction rate at x/L = 1. Distributed heat source with 25 mW is applied to 1/4 of the length of cavity. Blue: simplified radiation fin model. A nominal heat flux of 25 mW was applied at x/L = 0, and the standard tip condition was used, where radiation rate equals conduction rate at x/L = 1. Insert: FEA simulation of cavity temperature distribution. x = 0 indicates cavity center, heater power is applied to the left 1/4 section. Color map is the standard heat map, where white indicates hot.

Equations (13)

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ω = π q c n L = 2 π q Δ ν FSR
d T / d ω = [ ( α n + α L ) ω ] 1
G spacer ( f ) = 2 k B T f L 3 π 2 R 2 E s ϕ spacer
G sub ( f ) = 2 k B T f 1 σ 2 π 3 E m w 0 ϕ sub
G coat ( f ) = 2 ( 1 2 σ ) π ( 1 σ ) ϕ coat ϕ sub d w 0 G sub ( f )
ε 8 P c P s δ ν δ f = D PDH δ f
S e = 2 h c λ ( P ref )
S f = h c 3 8 1 L λ P c
ε P c P s Im [ t * ( ω ) ( t ( ω + Ω ) + t ( ω Ω ) ) ]
t ( ω ) = ( 1 R ) e i ω L / c 1 R e 2 i ω L / c
ε 8 P c P s δ v δ f = D dither δ f
G OPL ( f ) = 0 L η 0 α 2 4 π ϵ 0 2 w ( z ) ν 0 exp [ 2 π f w ( z ) v 0 ] d z
T res = η Q α = η λ 2 L α = η λ ν FWHM c α

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