Abstract

A mechanical force sensor coupled to two optical cavities is developed as a metrological tool. This system is used to generate a calibrated circulating optical power and to create a transfer standard for externally coupled optical power. The variability of the sensor as a transfer standard for optical power is less than 2%. The uncertainty in using the sensor to measure the circulating power inside the cavity is less than 3%. The force measured from the mechanical response of the sensor is compared to the force predicted from characterizing the optical spectrum of the cavity. These two forces are approximately 20% different. Potential sources for this disagreement are analyzed and discussed. The sensor is compact, portable, and can operate in ambient and vacuum environments. This device provides a pathway to novel nanonewton scale force and milliwatt scale laser power calibrations, enables direct measurement of the circulating power inside an optical cavity, and enhances the sensitivity of radiation pressure-based optical power transfer standards.

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References

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2017 (2)

2016 (2)

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

2014 (4)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

H. Takahashi, J. Morphew, F. Oručević, A. Noguchi, E. Kassa, and M. Keller, “Novel laser machining of optical fibers for long cavities with low birefringence,” Opt. Express 22(25), 31317–31328 (2014).
[Crossref] [PubMed]

K. Agatsuma, D. Friedrich, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, and S. Kawamura, “Precise measurement of laser power using an optomechanical system,” Opt. Express 22(2), 2013–2030 (2014).
[Crossref] [PubMed]

J. Melcher, J. Stirling, F. G. Cervantes, J. R. Pratt, and G. A. Shaw, “A self-calibrating optomechanical force sensor with femtonewton resolution,” Appl. Phys. Lett. 105(23), 233109 (2014).
[Crossref]

2013 (3)

P. A. Williams, J. A. Hadler, R. Lee, F. C. Maring, and J. H. Lehman, “Use of radiation pressure for measurement of high-power laser emission,” Opt. Lett. 38(20), 4248–4251 (2013).
[Crossref] [PubMed]

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

2008 (1)

2004 (1)

C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and torsional spring constants of atomic force microscope cantilevers,” Rev. Sci. Instrum. 75(6), 1988–1996 (2004).
[Crossref]

2000 (1)

E. D. Black, “An introduction to Pound–Drever–Hall laser frequency stabilization,” Am. J. Phys. 69(1), 79–87 (2000).
[Crossref]

1993 (1)

J. L. Hutter and J. Bechhoefer, “Calibration of atomic‐force microscope tips,” Rev. Sci. Instrum. 64(7), 1868–1873 (1993).
[Crossref]

1986 (1)

G. Binnig, C. F. Quate, and C. Gerber, “Atomic Force Microscope,” Phys. Rev. Lett. 56(9), 930–933 (1986).
[Crossref] [PubMed]

1901 (1)

E. F. Nichols and G. F. Hull, “A Preliminary Communication on the Pressure of Heat and Light Radiation,” Phys. Rev. Ser. I 13(5), 307–320 (1901).
[Crossref]

Agatsuma, K.

Alavi, S. K.

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

Alt, W.

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

Baek, B.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Ballmer, S.

Bechhoefer, J.

J. L. Hutter and J. Bechhoefer, “Calibration of atomic‐force microscope tips,” Rev. Sci. Instrum. 64(7), 1868–1873 (1993).
[Crossref]

Becker, C.

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

Bick, A.

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

Binnig, G.

G. Binnig, C. F. Quate, and C. Gerber, “Atomic Force Microscope,” Phys. Rev. Lett. 56(9), 930–933 (1986).
[Crossref] [PubMed]

Black, E. D.

E. D. Black, “An introduction to Pound–Drever–Hall laser frequency stabilization,” Am. J. Phys. 69(1), 79–87 (2000).
[Crossref]

Blatt, R.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Brandstätter, B.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Casabone, B.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Cervantes, F. G.

J. Melcher, J. Stirling, F. G. Cervantes, J. R. Pratt, and G. A. Shaw, “A self-calibrating optomechanical force sensor with femtonewton resolution,” Appl. Phys. Lett. 105(23), 233109 (2014).
[Crossref]

Christoph, P.

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

Cleveland, J. P.

C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and torsional spring constants of atomic force microscope cantilevers,” Rev. Sci. Instrum. 75(6), 1988–1996 (2004).
[Crossref]

DeSalvo, G.

Deutsch, C.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Feldman, A.

Friebe, K.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Friedrich, D.

Fröhlich, T.

Gallego, J.

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

Gerber, C.

G. Binnig, C. F. Quate, and C. Gerber, “Atomic Force Microscope,” Phys. Rev. Lett. 56(9), 930–933 (1986).
[Crossref] [PubMed]

Gerrits, T.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Ghosh, S.

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

Green, C. P.

C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and torsional spring constants of atomic force microscope cantilevers,” Rev. Sci. Instrum. 75(6), 1988–1996 (2004).
[Crossref]

Hadler, J.

Hadler, J. A.

Harrington, S.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Heinze, J.

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

Hellmig, O.

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

Heugel, S.

Hull, G. F.

E. F. Nichols and G. F. Hull, “A Preliminary Communication on the Pressure of Heat and Light Radiation,” Phys. Rev. Ser. I 13(5), 307–320 (1901).
[Crossref]

Hutter, J. L.

J. L. Hutter and J. Bechhoefer, “Calibration of atomic‐force microscope tips,” Rev. Sci. Instrum. 64(7), 1868–1873 (1993).
[Crossref]

Kassa, E.

Kawamura, S.

Keller, M.

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

Lee, R.

Lehman, J.

Lehman, J. H.

Lioe, H.

C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and torsional spring constants of atomic force microscope cantilevers,” Rev. Sci. Instrum. 75(6), 1988–1996 (2004).
[Crossref]

Lita, A. E.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Manske, E.

Maring, F.

Maring, F. C.

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

Marsili, F.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Martinez-Dorantes, M.

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

McClung, A.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Melcher, J.

J. Melcher, J. Stirling, F. G. Cervantes, J. R. Pratt, and G. A. Shaw, “A self-calibrating optomechanical force sensor with femtonewton resolution,” Appl. Phys. Lett. 105(23), 233109 (2014).
[Crossref]

Meschede, D.

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

Mirin, R. P.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Morphew, J.

Mueller, F.

Mulvaney, P.

C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and torsional spring constants of atomic force microscope cantilevers,” Rev. Sci. Instrum. 75(6), 1988–1996 (2004).
[Crossref]

Nam, S. W.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Nichols, E. F.

E. F. Nichols and G. F. Hull, “A Preliminary Communication on the Pressure of Heat and Light Radiation,” Phys. Rev. Ser. I 13(5), 307–320 (1901).
[Crossref]

Nishida, E.

Noguchi, A.

Northup, T. E.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Orucevic, F.

Pratt, J. R.

J. Melcher, J. Stirling, F. G. Cervantes, J. R. Pratt, and G. A. Shaw, “A self-calibrating optomechanical force sensor with femtonewton resolution,” Appl. Phys. Lett. 105(23), 233109 (2014).
[Crossref]

Proksch, R.

C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and torsional spring constants of atomic force microscope cantilevers,” Rev. Sci. Instrum. 75(6), 1988–1996 (2004).
[Crossref]

Quate, C. F.

G. Binnig, C. F. Quate, and C. Gerber, “Atomic Force Microscope,” Phys. Rev. Lett. 56(9), 930–933 (1986).
[Crossref] [PubMed]

Ratschbacher, L.

J. Gallego, S. Ghosh, S. K. Alavi, W. Alt, M. Martinez-Dorantes, D. Meschede, and L. Ratschbacher, “High-finesse fiber Fabry–Perot cavities: stabilization and mode matching analysis,” Appl. Phys. B 122(3), 47 (2016).
[Crossref]

Reichel, J.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Rogers, K.

Sader, J. E.

C. P. Green, H. Lioe, J. P. Cleveland, R. Proksch, P. Mulvaney, and J. E. Sader, “Normal and torsional spring constants of atomic force microscope cantilevers,” Rev. Sci. Instrum. 75(6), 1988–1996 (2004).
[Crossref]

Sakata, S.

Schmidt, P. O.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Schüppert, K.

B. Brandstätter, A. McClung, K. Schüppert, B. Casabone, K. Friebe, A. Stute, P. O. Schmidt, C. Deutsch, J. Reichel, R. Blatt, and T. E. Northup, “Integrated fiber-mirror ion trap for strong ion-cavity coupling,” Rev. Sci. Instrum. 84(12), 123104 (2013).
[Crossref] [PubMed]

Sengstock, K.

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

Shaw, G. A.

J. Melcher, J. Stirling, F. G. Cervantes, J. R. Pratt, and G. A. Shaw, “A self-calibrating optomechanical force sensor with femtonewton resolution,” Appl. Phys. Lett. 105(23), 233109 (2014).
[Crossref]

Shaw, M. D.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013).
[Crossref]

Simonds, B.

Spidell, M.

Staarmann, C.

A. Bick, C. Staarmann, P. Christoph, O. Hellmig, J. Heinze, K. Sengstock, and C. Becker, “The role of mode match in fiber cavities,” Rev. Sci. Instrum. 87(1), 013102 (2016).
[Crossref] [PubMed]

Stephens, M.

Stern, J. A.

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Certain commercial products are identified in this article in order to describe the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the products identified are necessarily the best available for the purpose.

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Figures (7)

Fig. 1
Fig. 1 a) Simplified schematic of optical train and force transducer. Red lines indicate the optical path. b) Final components in optical train before modulation cavity. Pin denotes optical power delivered to the fiber leading directly to the optical cavity after the vacuum feedthrough. By breaking the fiber connections and measuring power at P2, P3, PPC, and P2,3 we can calculate the losses lu and l2,3 and in turn calculate Pin as indicated in Eq. (13). c) Picture of second flexure, green false color is used to highlight the primary flexure structure. d) Pictures of optical cavities on a flexure.
Fig. 2
Fig. 2 a) Optical microscope image of the fiber end facet after laser machining. b) White light interferometric image of the end facet. This is used to reconstruct the surface of the laser machined fiber.
Fig. 3
Fig. 3 Flexure calibration. Both the added mass method and thermal method were used to calibrate the flexures; representative calibration data is shown in (a) and (b), respectively. The results of these calibration are shown in (c): k L = (84.7 ± 1.6) kN/m for the first flexure added mass method, k L = (63.2 ± 1.1) kN/m for the second flexure added mass method, k L = (76.3 ± 7.3) kN/m for the first flexure thermal method, and k L = (54.2 ± 7.0) kN/m for the second flexure thermal method.
Fig. 4
Fig. 4 Characterization of the fiber Fabry-Pérot cavity. a) long range wavelength sweep, b) short range wavelength sweep, c) FSR, d) finesse, e) ηdip (i.e dip parameter). The finesse, free spectral range, and the magnitude of the dip parameter was measured in order to calculate the cavity circulating power.
Fig. 5
Fig. 5 a) The response of the flexure as drive amplitude signal and drive frequency of the modulating input laser power is varied. b) A comparison of the force calculated from the flexure response (SHO) and the force calculated from the radiation pressure equation (Eq. (14)) and the fiber Fabry-Pérot cavity model (FFPC). c) A statistic comparison of the agreement between the two applied models.
Fig. 6
Fig. 6 Correlation between the measured cavity finesse for the two applied modes of force. The negative correlation on the second flexure could indicate that some of the observed disagreement between the two models is connected to unaccounted-for effects in our measurements of the cavity finesse.
Fig. 7
Fig. 7 Calibration curves for using the system as a power meter. The y-axis is the input power and the x-axis is the amplitude obtained by fitting a harmonic oscillation to the signal ( A sho ).

Tables (1)

Tables Icon

Table 1 A representative uncertainty analysis of one data point in one experiment on flexure 1. The flexure force, circulating power, and force ratio are combined standard uncertainties. Input variables have been averaged to reduce uncertainty in cases where we do not expect the measurand to vary. All uncertainties are reported as one standard deviation or one standard deviation of the mean as appropriate. The uncertainty in flexure stiffness arises from the uncertainty in fitting Eq. (4). The uncertainty in the input power is as described in the input power loss analysis. All other uncertainties are type A uncertainties.

Equations (14)

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A= A sho ( 1 ( ω ω 0 ) 2 ) 2 + ( ω Q ω 0 ) 2 ,
F sho = A sho k L ,
A= ν 0 2Δν( dV dλ ) V pdh ,
Δm= k L ( 1 ω 0 2 )  m e ,
1 2 k b T= 1 2 k L <x > 2 ,
P c = P in | η dip | | i 1 g rt e iϕ | 2 ,
η dip = α 2 t 1 2 ,
α= S Ψ cav * Ψ f dS,
  t 1 2 = P f,out / P f,in .
g rt = r 1 r 2 l cav ,
F= π g rt 1 g rt .
P r P in = | β η dip g rt e iϕ 1 g rt e iϕ | 2 , 
P in = ( P pc P 3 P 2,3 P 2 ) 1 2 ( P 2 P pd ) P pd '
  F fpc  = 2 P c c ,