Abstract

Force to frequency conversion (FFC) using the photoelastic effect inside a laser cavity is described. Analytic expressions are derived for scale factor, measurement errors, and laser dependent nonlinearity of appropriate instrument transducers. The influence of the laser pushing and pulling effects on linearity is calculated on the basis of the third-order saturation model. Our experiments with a modular test setup (633 nm) demonstrate FFC to be proportional to a high degree over almost 6 decades of input signal range. From 2 × 10−4 up to 80 N we observed the noise equivalent resolution of 10−4 N. The frequency response of our test setup was established from dc up to some kilohertz. FFC measurement range and resolution can be extended to 10−6 N or smaller values by applying improved laser stabilization and miniaturizing the cavity length and size of photoelastic material.

© 1989 Optical Society of America

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References

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  1. C. S. Draper, E. J. Frey, M. S. Sapuppo, P. K. Chapman, “Measurement of Small Specific Forces in Space,” Astronaut. Acta 11, 1–12 (1965).
  2. A. Abramovici, Z. Vager, M. Weksler, “Experimental Test of a Prototype Gravitational Radiation Detector Employing an Active Cavity Laser Sensor,” J. Phys. E 19, 182–188 (1986).
    [CrossRef]
  3. W. B. Spillman, D. H. McMahon, “Multimode Fiber-Optic Hydrophone Based on the Photoelastic Effect,” Appl. Opt. 21, 3511–3514 (1982).
    [CrossRef] [PubMed]
  4. A. J. Barlow, D. N. Payne, “The Stress-Optic Effect in Optical Fibers,” IEEE J. Quantum Electron. QE-19, 834–839 (1983).
    [CrossRef]
  5. G. Martens, “Measurement of Pressure by Photoelastic Effects,” Sensors Actuators Vol. 6, 181–190 (1984).
    [CrossRef]
  6. G. Martens, “Photoelastic Fiber-Optical Sensor for Measurement of Force and Pressure,” Tech. Messen 53, No. 9, 331–338 (1986).
  7. J. C. Canit, J. Badoz, “Photoelastic Modulator for Polarimetry and Ellipsometry,” Appl. Opt. 23, 2861–2862 (1984).
    [CrossRef] [PubMed]
  8. J. A. Abate, St. D. Jacobs, “Spatially Resolved Characterization of Thin Films,” LLE Rev. 10, 3–6 (1982).
  9. A. J. Barlow, “Optical-Fiber Birefringence Measurement Using a Photo-Elastic Modulator,” IEEE/OSA J. Lightwave Technol. LT-3, 135–145 (1985).
    [CrossRef]
  10. W. B. Spillmann, “Multimode Fiber-Optic Accelerometer Based on the Photoelastic Effect,” Appl. Opt. 21, 2653–2655 (1982).
    [CrossRef]
  11. S. Tai, K. Kyuma, M. Nunoshita, “Fiber-Optic Acceleration Sensor Based on the Photoelastic Effect,” Appl. Opt. 22, 1771–1774 (1983).
    [CrossRef] [PubMed]
  12. E. D. T. Jacobs, W. L. Zingery, “Accelerometer,” U.S. Patent3,517,560 (1970).
  13. W. Holzapfel, “Laser in Measurement and Navigation,” Laser Elektro-Opt. 5, 3–7 (1973).
  14. T. J. Hutchings, W. L. Zingery, “Laser Accelerometer, U.S. Patent3,800,594 (1974).
  15. G. D. Babcock, “Laser Accelerometer,” U.S. Patent4,048,859 (Oct.1975).
  16. W. Holzapfel, “Inertial Acceleration Measurement by Laser Methods and Systems,” BMFT-FB W 75-30 (12) (1975).
  17. W. Holzapfel, “Acceleration and Force Measurement System,” Patent DE2633178C3 (1981).
  18. W. Holzapfel, U. Riss, “Computer-Based High Resolution Transmission Ellipsometry,” Appl. Opt. 26, 145–153 (1987).
    [CrossRef] [PubMed]
  19. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  20. D. Polder, W. van Haeringen, “The Effect of Saturation on the Ellipticity of Modes in Gas Lasers,” Phys. Lett. 19, 380–381 (1965).
    [CrossRef]
  21. W. van Haeringen, “Polarization Properties of a Single Mode Operating Gas Laser in a Small Axial Magnetic Field,” Phys. Rev. 158, 256–272 (1967).
    [CrossRef]
  22. R. A. Keijser, “Polarization Properties of Internal Mirror He–Ne Lasers in a Strong Transverse Magnetic Field,” Opt. Commun. 23, 194–198 (1977).
    [CrossRef]
  23. D. Lenstra, G. C. Herman, “Saturation Induced Polarization Preferences in Two-Mode Oscillation Gas Lasers,” Phys. C 95, 405–411 (1978).
  24. W. M. Doyle, M. B. White, “Properties of an Anisotropic Fabry-Perot Resonator,” J. Opt. Soc. Am. 55, 1221–1225 (1965).
    [CrossRef]
  25. H. de Lang, “Polarization Properties of Optical Resonators Passive and Active,” Philips Res. Rep. Supplement No. 8, (1967), pp. 4–20.
  26. W. E. Lamb, “Theory of an Optical Maser,” Phys. Rev. A 134, A1429–A1450 (1964).
  27. M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, London, 1974), pp. 96–170.
  28. A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 214–328.
  29. R. A. McFarlane, “Frequency Pushing and Frequency Pulling in a He-Ne Gas Optical Maser,” Phys. Rev. A 135, A543–A550 (1964).
  30. R. L. Fork, M. A. Pollack, “Mode Competition and Collisions Effects in Gaseous Optical Masers,” Phys. Rev. A 139, A1408–A1414 (1965).
  31. P. W. Smith, “Linewidth and Saturation Parameters for the 6328-Å Transition in a He–Ne Laser,” J. Appl. Phys. 37, 2089–2093 (1966).
    [CrossRef]
  32. R. Balhorn, H. Kunzmann, F. Lebowsky, “Frequency Stabilization of Internal-Mirror Helium–Neon Lasers,” Appl. Opt. 11, 742–744 (1972).
    [CrossRef] [PubMed]
  33. H. Wolf, Spannungsoptik, Bd. 1 (Berlin, New York, 1976), pp. 77–102.
  34. B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).
  35. M. Abramowitz, T. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, DC, 1964), pp. 297–330.

1987 (1)

1986 (2)

G. Martens, “Photoelastic Fiber-Optical Sensor for Measurement of Force and Pressure,” Tech. Messen 53, No. 9, 331–338 (1986).

A. Abramovici, Z. Vager, M. Weksler, “Experimental Test of a Prototype Gravitational Radiation Detector Employing an Active Cavity Laser Sensor,” J. Phys. E 19, 182–188 (1986).
[CrossRef]

1985 (1)

A. J. Barlow, “Optical-Fiber Birefringence Measurement Using a Photo-Elastic Modulator,” IEEE/OSA J. Lightwave Technol. LT-3, 135–145 (1985).
[CrossRef]

1984 (2)

G. Martens, “Measurement of Pressure by Photoelastic Effects,” Sensors Actuators Vol. 6, 181–190 (1984).
[CrossRef]

J. C. Canit, J. Badoz, “Photoelastic Modulator for Polarimetry and Ellipsometry,” Appl. Opt. 23, 2861–2862 (1984).
[CrossRef] [PubMed]

1983 (2)

S. Tai, K. Kyuma, M. Nunoshita, “Fiber-Optic Acceleration Sensor Based on the Photoelastic Effect,” Appl. Opt. 22, 1771–1774 (1983).
[CrossRef] [PubMed]

A. J. Barlow, D. N. Payne, “The Stress-Optic Effect in Optical Fibers,” IEEE J. Quantum Electron. QE-19, 834–839 (1983).
[CrossRef]

1982 (3)

1978 (1)

D. Lenstra, G. C. Herman, “Saturation Induced Polarization Preferences in Two-Mode Oscillation Gas Lasers,” Phys. C 95, 405–411 (1978).

1977 (1)

R. A. Keijser, “Polarization Properties of Internal Mirror He–Ne Lasers in a Strong Transverse Magnetic Field,” Opt. Commun. 23, 194–198 (1977).
[CrossRef]

1975 (1)

W. Holzapfel, “Inertial Acceleration Measurement by Laser Methods and Systems,” BMFT-FB W 75-30 (12) (1975).

1973 (1)

W. Holzapfel, “Laser in Measurement and Navigation,” Laser Elektro-Opt. 5, 3–7 (1973).

1972 (1)

1967 (2)

W. van Haeringen, “Polarization Properties of a Single Mode Operating Gas Laser in a Small Axial Magnetic Field,” Phys. Rev. 158, 256–272 (1967).
[CrossRef]

H. de Lang, “Polarization Properties of Optical Resonators Passive and Active,” Philips Res. Rep. Supplement No. 8, (1967), pp. 4–20.

1966 (1)

P. W. Smith, “Linewidth and Saturation Parameters for the 6328-Å Transition in a He–Ne Laser,” J. Appl. Phys. 37, 2089–2093 (1966).
[CrossRef]

1965 (4)

R. L. Fork, M. A. Pollack, “Mode Competition and Collisions Effects in Gaseous Optical Masers,” Phys. Rev. A 139, A1408–A1414 (1965).

W. M. Doyle, M. B. White, “Properties of an Anisotropic Fabry-Perot Resonator,” J. Opt. Soc. Am. 55, 1221–1225 (1965).
[CrossRef]

D. Polder, W. van Haeringen, “The Effect of Saturation on the Ellipticity of Modes in Gas Lasers,” Phys. Lett. 19, 380–381 (1965).
[CrossRef]

C. S. Draper, E. J. Frey, M. S. Sapuppo, P. K. Chapman, “Measurement of Small Specific Forces in Space,” Astronaut. Acta 11, 1–12 (1965).

1964 (2)

W. E. Lamb, “Theory of an Optical Maser,” Phys. Rev. A 134, A1429–A1450 (1964).

R. A. McFarlane, “Frequency Pushing and Frequency Pulling in a He-Ne Gas Optical Maser,” Phys. Rev. A 135, A543–A550 (1964).

Abate, J. A.

J. A. Abate, St. D. Jacobs, “Spatially Resolved Characterization of Thin Films,” LLE Rev. 10, 3–6 (1982).

Abramovici, A.

A. Abramovici, Z. Vager, M. Weksler, “Experimental Test of a Prototype Gravitational Radiation Detector Employing an Active Cavity Laser Sensor,” J. Phys. E 19, 182–188 (1986).
[CrossRef]

Abramowitz, M.

M. Abramowitz, T. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, DC, 1964), pp. 297–330.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Babcock, G. D.

G. D. Babcock, “Laser Accelerometer,” U.S. Patent4,048,859 (Oct.1975).

Badoz, J.

Balhorn, R.

Barlow, A. J.

A. J. Barlow, “Optical-Fiber Birefringence Measurement Using a Photo-Elastic Modulator,” IEEE/OSA J. Lightwave Technol. LT-3, 135–145 (1985).
[CrossRef]

A. J. Barlow, D. N. Payne, “The Stress-Optic Effect in Optical Fibers,” IEEE J. Quantum Electron. QE-19, 834–839 (1983).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Canit, J. C.

Chapman, P. K.

C. S. Draper, E. J. Frey, M. S. Sapuppo, P. K. Chapman, “Measurement of Small Specific Forces in Space,” Astronaut. Acta 11, 1–12 (1965).

Conte, S. D.

B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

de Lang, H.

H. de Lang, “Polarization Properties of Optical Resonators Passive and Active,” Philips Res. Rep. Supplement No. 8, (1967), pp. 4–20.

Doyle, W. M.

Draper, C. S.

C. S. Draper, E. J. Frey, M. S. Sapuppo, P. K. Chapman, “Measurement of Small Specific Forces in Space,” Astronaut. Acta 11, 1–12 (1965).

Dunn, M. H.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 214–328.

Fork, R. L.

R. L. Fork, M. A. Pollack, “Mode Competition and Collisions Effects in Gaseous Optical Masers,” Phys. Rev. A 139, A1408–A1414 (1965).

Frey, E. J.

C. S. Draper, E. J. Frey, M. S. Sapuppo, P. K. Chapman, “Measurement of Small Specific Forces in Space,” Astronaut. Acta 11, 1–12 (1965).

Fried, B. D.

B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

Herman, G. C.

D. Lenstra, G. C. Herman, “Saturation Induced Polarization Preferences in Two-Mode Oscillation Gas Lasers,” Phys. C 95, 405–411 (1978).

Holzapfel, W.

W. Holzapfel, U. Riss, “Computer-Based High Resolution Transmission Ellipsometry,” Appl. Opt. 26, 145–153 (1987).
[CrossRef] [PubMed]

W. Holzapfel, “Inertial Acceleration Measurement by Laser Methods and Systems,” BMFT-FB W 75-30 (12) (1975).

W. Holzapfel, “Laser in Measurement and Navigation,” Laser Elektro-Opt. 5, 3–7 (1973).

W. Holzapfel, “Acceleration and Force Measurement System,” Patent DE2633178C3 (1981).

Hutchings, T. J.

T. J. Hutchings, W. L. Zingery, “Laser Accelerometer, U.S. Patent3,800,594 (1974).

Jacobs, E. D. T.

E. D. T. Jacobs, W. L. Zingery, “Accelerometer,” U.S. Patent3,517,560 (1970).

Jacobs, St. D.

J. A. Abate, St. D. Jacobs, “Spatially Resolved Characterization of Thin Films,” LLE Rev. 10, 3–6 (1982).

Keijser, R. A.

R. A. Keijser, “Polarization Properties of Internal Mirror He–Ne Lasers in a Strong Transverse Magnetic Field,” Opt. Commun. 23, 194–198 (1977).
[CrossRef]

Kunzmann, H.

Kyuma, K.

Lamb, W. E.

W. E. Lamb, “Theory of an Optical Maser,” Phys. Rev. A 134, A1429–A1450 (1964).

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, London, 1974), pp. 96–170.

Lebowsky, F.

Lenstra, D.

D. Lenstra, G. C. Herman, “Saturation Induced Polarization Preferences in Two-Mode Oscillation Gas Lasers,” Phys. C 95, 405–411 (1978).

Maitland, A.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 214–328.

Martens, G.

G. Martens, “Photoelastic Fiber-Optical Sensor for Measurement of Force and Pressure,” Tech. Messen 53, No. 9, 331–338 (1986).

G. Martens, “Measurement of Pressure by Photoelastic Effects,” Sensors Actuators Vol. 6, 181–190 (1984).
[CrossRef]

McFarlane, R. A.

R. A. McFarlane, “Frequency Pushing and Frequency Pulling in a He-Ne Gas Optical Maser,” Phys. Rev. A 135, A543–A550 (1964).

McMahon, D. H.

Nunoshita, M.

Payne, D. N.

A. J. Barlow, D. N. Payne, “The Stress-Optic Effect in Optical Fibers,” IEEE J. Quantum Electron. QE-19, 834–839 (1983).
[CrossRef]

Polder, D.

D. Polder, W. van Haeringen, “The Effect of Saturation on the Ellipticity of Modes in Gas Lasers,” Phys. Lett. 19, 380–381 (1965).
[CrossRef]

Pollack, M. A.

R. L. Fork, M. A. Pollack, “Mode Competition and Collisions Effects in Gaseous Optical Masers,” Phys. Rev. A 139, A1408–A1414 (1965).

Riss, U.

Sapuppo, M. S.

C. S. Draper, E. J. Frey, M. S. Sapuppo, P. K. Chapman, “Measurement of Small Specific Forces in Space,” Astronaut. Acta 11, 1–12 (1965).

Sargent, M.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, London, 1974), pp. 96–170.

Scully, M. O.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, London, 1974), pp. 96–170.

Smith, P. W.

P. W. Smith, “Linewidth and Saturation Parameters for the 6328-Å Transition in a He–Ne Laser,” J. Appl. Phys. 37, 2089–2093 (1966).
[CrossRef]

Spillman, W. B.

Spillmann, W. B.

Stegun, T. A.

M. Abramowitz, T. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, DC, 1964), pp. 297–330.

Tai, S.

Vager, Z.

A. Abramovici, Z. Vager, M. Weksler, “Experimental Test of a Prototype Gravitational Radiation Detector Employing an Active Cavity Laser Sensor,” J. Phys. E 19, 182–188 (1986).
[CrossRef]

van Haeringen, W.

W. van Haeringen, “Polarization Properties of a Single Mode Operating Gas Laser in a Small Axial Magnetic Field,” Phys. Rev. 158, 256–272 (1967).
[CrossRef]

D. Polder, W. van Haeringen, “The Effect of Saturation on the Ellipticity of Modes in Gas Lasers,” Phys. Lett. 19, 380–381 (1965).
[CrossRef]

Weksler, M.

A. Abramovici, Z. Vager, M. Weksler, “Experimental Test of a Prototype Gravitational Radiation Detector Employing an Active Cavity Laser Sensor,” J. Phys. E 19, 182–188 (1986).
[CrossRef]

White, M. B.

Wolf, H.

H. Wolf, Spannungsoptik, Bd. 1 (Berlin, New York, 1976), pp. 77–102.

Zingery, W. L.

T. J. Hutchings, W. L. Zingery, “Laser Accelerometer, U.S. Patent3,800,594 (1974).

E. D. T. Jacobs, W. L. Zingery, “Accelerometer,” U.S. Patent3,517,560 (1970).

Appl. Opt. (6)

Astronaut. Acta (1)

C. S. Draper, E. J. Frey, M. S. Sapuppo, P. K. Chapman, “Measurement of Small Specific Forces in Space,” Astronaut. Acta 11, 1–12 (1965).

BMFT-FB W 75-30 (1)

W. Holzapfel, “Inertial Acceleration Measurement by Laser Methods and Systems,” BMFT-FB W 75-30 (12) (1975).

IEEE J. Quantum Electron. (1)

A. J. Barlow, D. N. Payne, “The Stress-Optic Effect in Optical Fibers,” IEEE J. Quantum Electron. QE-19, 834–839 (1983).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (1)

A. J. Barlow, “Optical-Fiber Birefringence Measurement Using a Photo-Elastic Modulator,” IEEE/OSA J. Lightwave Technol. LT-3, 135–145 (1985).
[CrossRef]

J. Appl. Phys. (1)

P. W. Smith, “Linewidth and Saturation Parameters for the 6328-Å Transition in a He–Ne Laser,” J. Appl. Phys. 37, 2089–2093 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. E (1)

A. Abramovici, Z. Vager, M. Weksler, “Experimental Test of a Prototype Gravitational Radiation Detector Employing an Active Cavity Laser Sensor,” J. Phys. E 19, 182–188 (1986).
[CrossRef]

Laser Elektro-Opt. (1)

W. Holzapfel, “Laser in Measurement and Navigation,” Laser Elektro-Opt. 5, 3–7 (1973).

LLE Rev. (1)

J. A. Abate, St. D. Jacobs, “Spatially Resolved Characterization of Thin Films,” LLE Rev. 10, 3–6 (1982).

Opt. Commun. (1)

R. A. Keijser, “Polarization Properties of Internal Mirror He–Ne Lasers in a Strong Transverse Magnetic Field,” Opt. Commun. 23, 194–198 (1977).
[CrossRef]

Philips Res. Rep. (1)

H. de Lang, “Polarization Properties of Optical Resonators Passive and Active,” Philips Res. Rep. Supplement No. 8, (1967), pp. 4–20.

Phys. C (1)

D. Lenstra, G. C. Herman, “Saturation Induced Polarization Preferences in Two-Mode Oscillation Gas Lasers,” Phys. C 95, 405–411 (1978).

Phys. Lett. (1)

D. Polder, W. van Haeringen, “The Effect of Saturation on the Ellipticity of Modes in Gas Lasers,” Phys. Lett. 19, 380–381 (1965).
[CrossRef]

Phys. Rev. (1)

W. van Haeringen, “Polarization Properties of a Single Mode Operating Gas Laser in a Small Axial Magnetic Field,” Phys. Rev. 158, 256–272 (1967).
[CrossRef]

Phys. Rev. A (3)

W. E. Lamb, “Theory of an Optical Maser,” Phys. Rev. A 134, A1429–A1450 (1964).

R. A. McFarlane, “Frequency Pushing and Frequency Pulling in a He-Ne Gas Optical Maser,” Phys. Rev. A 135, A543–A550 (1964).

R. L. Fork, M. A. Pollack, “Mode Competition and Collisions Effects in Gaseous Optical Masers,” Phys. Rev. A 139, A1408–A1414 (1965).

Sensors Actuators (1)

G. Martens, “Measurement of Pressure by Photoelastic Effects,” Sensors Actuators Vol. 6, 181–190 (1984).
[CrossRef]

Tech. Messen (1)

G. Martens, “Photoelastic Fiber-Optical Sensor for Measurement of Force and Pressure,” Tech. Messen 53, No. 9, 331–338 (1986).

Other (10)

E. D. T. Jacobs, W. L. Zingery, “Accelerometer,” U.S. Patent3,517,560 (1970).

T. J. Hutchings, W. L. Zingery, “Laser Accelerometer, U.S. Patent3,800,594 (1974).

G. D. Babcock, “Laser Accelerometer,” U.S. Patent4,048,859 (Oct.1975).

W. Holzapfel, “Acceleration and Force Measurement System,” Patent DE2633178C3 (1981).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, London, 1974), pp. 96–170.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 214–328.

H. Wolf, Spannungsoptik, Bd. 1 (Berlin, New York, 1976), pp. 77–102.

B. D. Fried, S. D. Conte, The Plasma Dispersion Function (Academic, New York, 1961).

M. Abramowitz, T. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, DC, 1964), pp. 297–330.

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Figures (9)

Fig. 1
Fig. 1

Principal design of the laser FFC system. F, force (input signal); Tp, Jones matrix of pth resonator element; ν1,ν2, optical frequencies orthogonally polarized.

Fig. 2
Fig. 2

Force induced frequency variations of orthogonally polarized mode frequencies ν1,ν2 of a He–Ne laser: (a) fixed frequency stabilization method; (b) symmetrical mode tuning stabilization method. ν0, line center frequency; νi, optical frequency; f0, beat frequency without force (offset frequency); Δf + δf, change of beat frequency.

Fig. 3
Fig. 3

FFC system: a general schematic block diagram of the FFC: Fm, force acting on the PEM; cL, stiffness of the load system; νi, laser mode; νref, reference frequency; BS, beam splitter; P, polarizer; PD, photodiode; U, control voltage.

Fig. 4
Fig. 4

FFC system: signal processing of the force input Fm to the frequency output Δf. cPEM, stiffness of the PEM; δFk, disturbing forces; D0, initial optical resonator length; Dc, resonator length control signal; νci, cavity resonance frequency; δνi,k, deviations from cavity resonance frequency νci; N, laser nonlinearities; other abbreviations as in Fig. 3.

Fig. 5
Fig. 5

Block diagram of laser nonlinearity N showing the influence on beat frequency f as a function of symmetrically tuned cavity resonances νci.

Fig. 6
Fig. 6

Calculated nonlinear deviation δfNL of beat frequency f caused by laser nonlinearities; η, normalized gain coefficient.

Fig. 7
Fig. 7

Schematic block diagram of the FFC experimental setup: PZT, piezoceramic drive; PID, proportional integral derivate control.

Fig. 8
Fig. 8

Time trace of beat frequency change Δf caused by a series of input step signals, force step ΔF ≅ 1 N. FSR = 358 MHz; PEM1: BK7 (C0 = 2.79 × 10−6 mm2/N), size, Ø = 10 mm, L0 = 5 mm; AR coated with a reflectivity of < 0.1%/side.

Fig. 9
Fig. 9

Static characteristics of the FFC experimental setup. Different scale factors E1 and E2 are obtained by changing the FSR value: E1, FSR1 = 108 MHz, PEM1; E2, FSR2 = 358 MHz, PEM1; E3, FSR3 = 358 MHz, PEM 2: material BK7; size, Ø = 3 mm, L0 = 5 mm; E4, FSR4 = 1 GHz, PEM 3: material KzF6, C0 = 4.05 × 10−6 mm2/N; size, W = 1 mm, L0 = 3 mm. All elements AR coated, reflectivity, ≤0.1%/side. □, lower limit due to the frequency instability of the experimental setup; ▲, lower limit, calculated from the measured frequency instability of a modified laser system.

Equations (38)

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T res = ( p = 1 n T p T ) ( p = n 1 T p ) = [ exp ( j Δ / 2 ) 0 0 exp ( j Δ / 2 ) ] ,
Δ = 2 p = 1 n Δ p ,
Δ σ = 1 W L 0 k s F .
Δ L = λ 2 π Δ ind = C 0 L 0 Δ σ .
f = ν 2 ν 1 = f 0 + Δ f + δ f ,
f 0 + Δ f = δ f = ν c 2 ν c 1 , k = 1 m δ f k = k = 1 m ( δ ν 2 , k δ ν 1 , k ) ,
Δ f = 2 ( FSR ) λ ( C 0 W ) k s F = E F ,
δ σ d = ( δ F 1 ) k s L 0 W = h [ α a Δ T a α b Δ T b ] c L · k s L 0 W ,
δ f 1 = ( 2 C 0 ) L 0 λ ( FSR · δ σ d ) .
δ f 2 = E ( 1 cos 2 β ) F .
k = 3 6 δ f k = [ E C 0 ( d C 0 ) + E D 0 ( d D 0 ) + E W ( d W ) + E λ ( d λ ) ] F .
Δ FSR = FSR 2 c Δ D
δ F = ± 1 E δ f .
f = ν 2 ν 1 = ν c 2 ν c 1 + δ ν 2 * δ ν 1 * .
δ ν i * δ ν i 7 = δ ν i 1 * + δ ν i 2 * + δ ν i 3 * i = 1 , 2 .
δ ν i *
δ ν i 1 *
δ ν 2 *
δ ν 3 *
δ ν 1 * = δ ν 11 * + δ ν 12 * + δ ν 13 * ,
δ ν 2 * = δ ν 21 * + δ ν 22 * + δ ν 23 * ,
δ ν i 1 * S i ( Δ ν i ) = χ 2 ( η i 1 / i 2 ) Z r ( Δ ν i ) Z i ( 0 ) ,
δ ν i 2 * R i ( Δ ν i , I i ) = ρ i I i ,
δ ν i 3 * T i ( Δ ν 1 , Δ ν 2 , I i ) = τ i i + 1 , i + 1 i I i + 1 , i .
Δ ν i = ν i ν 0 , i = 1 , 2
Z r ( Δ v i , A ) = 2 D ( Δ v i ) + 2 Δ v i k u A exp ( Δ v i 2 k u 2 ) π ,
Z i ( Δ v i , A ) = exp ( Δ v i 2 k u 2 ) π + 4 A Δ v i k u D ( Δ v i ) 2 A
D ( Φ ) = exp ( Φ 2 ) 0 Φ exp ( y 2 ) d y
A = γ s k u ; γ s = γ + D s · p ; γ = ( γ 1 + γ 2 ) 2 + D h p .
I i = I i * ( Δ ν 1 , Δ ν 2 , k 1 , k 2 , k b 1 , k b 2 ) U i ( Δ ν 1 , Δ ν 2 , k 1 , k 2 , k b 1 , k b 2 ) ,
I i * = α i β i 1 ( θ 12 θ 21 β 1 β 2 ) ; U i = I i * θ 12 / 21 β i .
α i = χ 2 η 11 , 12 [ Z i ( Δ ν i ) Z i ( 0 ) 1 η 11 / 12 ] ,
β i = F ( η 1 i ) [ 1 + k b i γ s 2 L ( Δ ν i ) ] ,
θ 12 / 21 = F ( η 1 i ) { [ 1 + γ s 2 · L ( X 2 ) ] k i + γ L ( X 2 ) [ γ s ( γ 1 γ 1 2 + X 2 + γ 2 γ 2 2 + X 2 ) X 2 ( X γ 1 2 + X 2 + X γ 2 2 + X 2 ) ] + η 21 / 22 γ L ( Δ ν 1 / 2 ) [ γ s ( γ 1 γ 1 2 + X 2 + γ 2 γ 2 2 + X 2 ) ± Δ ν 1 / 2 ( X γ 1 2 + X 2 + X γ 2 2 + X 2 ) ] } .
ρ i = F ( η 1 i ) γ s Δ ν i L ( Δ ν i ) .
τ 12 / 21 = F ( η 1 i ) { ± γ s X 2 L ( X 2 ) ± γ L ( X 2 ) [ X 2 · ( γ 1 γ 1 2 + X 2 + γ 2 γ 2 2 + X 2 ) + γ s ( X γ 1 2 + X 2 + X γ 2 2 + X 2 ) ] + η 21 / 22 γ L ( Δ ν 1 / 2 ) [ Δ ν 1 / 2 ( γ 1 γ 1 2 + X 2 + γ 2 γ 2 2 + X 2 ) ± γ s ( X γ 1 2 + X 2 + X γ 2 2 + X 2 ) ] }
F ( η 1 i ) = η 1 i χ π F dipole γ 1 γ 2 γ rel Z i ( 0 ) ; L ( y ) = 1 γ s 2 + y 2 ,
γ = γ 1 γ 2 γ rel 2 ; γ rel = 2 γ s ( γ 1 + γ 2 ) ; X = | Δ ν 1 | + | Δ ν 2 | .

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