Abstract

Accurate and traceable length metrology is employed by laser frequency stabilization. This paper describes a laser frequency stabilzation technique as a secondary standard with a fractional frequency stability of 5.2×10-10 with 2 mW of power, suitable for practical applications. The feedback stabilization is driven by an intrinsic mixed mode signal, caused by nonlinear optical phenomena with adjacent modes. The mixed mode signals are described theoretically and experimentally verified.

© 2010 Optical Society of America

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References

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  1. T. J. Quin, "Mise en Pratique of the Definition of the Metre (1992)," Metrologia 30, 523-541 (1994).
    [CrossRef]
  2. D. A. Jennings, C. R. Pollack, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, J. L. Hall, and H. P. Layer, "Direct frequency measurement of the I2-stabilized He-Ne 473-THz (633-nm) laser," Opt. Lett. 8, 136-138 (1983).
    [CrossRef] [PubMed]
  3. T. H. Yoon, J. Ye, J. L. Hall, and J. -M. Chartier, "Absolute frequency measurement of the iodine-stabilized He-Ne laser at 633 nm," Appl. Phys. B 72, 221-226 (2001).
  4. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
    [CrossRef] [PubMed]
  5. Th. Udem, R. Holzwarth, and T. W. Hansch, "Optical frequency metrology," Nature 416, 233-237 (2002).
    [CrossRef] [PubMed]
  6. J. Lawall, J. M. Pedulla, and Y. L. Coq, "Ultrastable laser array at 633 nm for real-time dimensional metrology," Rev. Sci. Instrum. 72, 2879-2888 (2001).
    [CrossRef]
  7. F. C. Demarest, "High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics," Meas. Sci. Technol. 9, 1024-1030 (1998).
    [CrossRef]
  8. B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, "Validation of a single fibre-fed heterodyne laser interferometer with nanometre uncertainty," Precis. Eng. 29, 229-236 (2005).
    [CrossRef]
  9. T. Baer, F. V. Kowalski, and J. L. Hall, "Frequency stabilization of a 0.633-μm He-Ne longitudinal Zeeman laser," Appl. Opt. 19, 3173-3177 (1980).
    [CrossRef] [PubMed]
  10. R. Balhorn, H. Kunzmann, and F. Lebowsky, "Frequency Stabilization of Internal-Mirror Helium-Neon Lasers," Appl. Opt. 11, 742-744 (1972)
    [CrossRef] [PubMed]
  11. H. S. Suh, T. H. Yoon, M. S. Chung, and O. S. Choi, "Frequency and power stabilization of a three longitudinal mode He-Ne laser using secondary beat frequency" Appl. Phys. Let. 63, 2027-2029 (1993)
    [CrossRef]
  12. S. Yokoyama, T. Araki, and N. Suzuki, "Intermode beat stabilized laser with frequency pulling," Appl. Opt. 33, 358-363 (1994)
    [CrossRef] [PubMed]
  13. J. Y. Yeom and T. H. Yoon, "Three-longitudinal-mode He-Ne laser frequency stabilized at 633 nm by thermal phase locking of the secondary beat frequency" Appl. Opt. 44, 266-270 (2005)
    [CrossRef] [PubMed]
  14. W. E. Lamb, "Theory of an optical maser," Phys. Rev. 134A1429-A1450 (1964)
    [CrossRef]
  15. M. D. Sayers and L. Allen, "Amplitude, competition, self-locking, beat frequency, and time dependent in a threemode gas laser," Phys. Rev. A 1, 1730-1746 (1970)
    [CrossRef]
  16. H. Dekker, "Theory of self-locking phenomena in the pressure broadened three-mode He-Ne laser," Appl. Phys. 4, 257-263 (1974)
    [CrossRef]
  17. T. Yokoyama, T. Araki, S. Yokoyama, and N. Suzuki, "A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser" Meas. Sci. Technol. 12, 157-162 (2001)
    [CrossRef]

2005 (2)

B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, "Validation of a single fibre-fed heterodyne laser interferometer with nanometre uncertainty," Precis. Eng. 29, 229-236 (2005).
[CrossRef]

J. Y. Yeom and T. H. Yoon, "Three-longitudinal-mode He-Ne laser frequency stabilized at 633 nm by thermal phase locking of the secondary beat frequency" Appl. Opt. 44, 266-270 (2005)
[CrossRef] [PubMed]

2002 (1)

Th. Udem, R. Holzwarth, and T. W. Hansch, "Optical frequency metrology," Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

2001 (3)

J. Lawall, J. M. Pedulla, and Y. L. Coq, "Ultrastable laser array at 633 nm for real-time dimensional metrology," Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

T. H. Yoon, J. Ye, J. L. Hall, and J. -M. Chartier, "Absolute frequency measurement of the iodine-stabilized He-Ne laser at 633 nm," Appl. Phys. B 72, 221-226 (2001).

T. Yokoyama, T. Araki, S. Yokoyama, and N. Suzuki, "A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser" Meas. Sci. Technol. 12, 157-162 (2001)
[CrossRef]

2000 (1)

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

1998 (1)

F. C. Demarest, "High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics," Meas. Sci. Technol. 9, 1024-1030 (1998).
[CrossRef]

1994 (2)

1993 (1)

H. S. Suh, T. H. Yoon, M. S. Chung, and O. S. Choi, "Frequency and power stabilization of a three longitudinal mode He-Ne laser using secondary beat frequency" Appl. Phys. Let. 63, 2027-2029 (1993)
[CrossRef]

1983 (1)

1980 (1)

1974 (1)

H. Dekker, "Theory of self-locking phenomena in the pressure broadened three-mode He-Ne laser," Appl. Phys. 4, 257-263 (1974)
[CrossRef]

1972 (1)

1970 (1)

M. D. Sayers and L. Allen, "Amplitude, competition, self-locking, beat frequency, and time dependent in a threemode gas laser," Phys. Rev. A 1, 1730-1746 (1970)
[CrossRef]

1964 (1)

W. E. Lamb, "Theory of an optical maser," Phys. Rev. 134A1429-A1450 (1964)
[CrossRef]

Allen, L.

M. D. Sayers and L. Allen, "Amplitude, competition, self-locking, beat frequency, and time dependent in a threemode gas laser," Phys. Rev. A 1, 1730-1746 (1970)
[CrossRef]

Araki, T.

T. Yokoyama, T. Araki, S. Yokoyama, and N. Suzuki, "A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser" Meas. Sci. Technol. 12, 157-162 (2001)
[CrossRef]

S. Yokoyama, T. Araki, and N. Suzuki, "Intermode beat stabilized laser with frequency pulling," Appl. Opt. 33, 358-363 (1994)
[CrossRef] [PubMed]

Baer, T.

Balhorn, R.

Chartier, J. -M.

T. H. Yoon, J. Ye, J. L. Hall, and J. -M. Chartier, "Absolute frequency measurement of the iodine-stabilized He-Ne laser at 633 nm," Appl. Phys. B 72, 221-226 (2001).

Choi, O. S.

H. S. Suh, T. H. Yoon, M. S. Chung, and O. S. Choi, "Frequency and power stabilization of a three longitudinal mode He-Ne laser using secondary beat frequency" Appl. Phys. Let. 63, 2027-2029 (1993)
[CrossRef]

Chung, M. S.

H. S. Suh, T. H. Yoon, M. S. Chung, and O. S. Choi, "Frequency and power stabilization of a three longitudinal mode He-Ne laser using secondary beat frequency" Appl. Phys. Let. 63, 2027-2029 (1993)
[CrossRef]

Coq, Y. L.

J. Lawall, J. M. Pedulla, and Y. L. Coq, "Ultrastable laser array at 633 nm for real-time dimensional metrology," Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

Cosijns, S. J. A. G.

B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, "Validation of a single fibre-fed heterodyne laser interferometer with nanometre uncertainty," Precis. Eng. 29, 229-236 (2005).
[CrossRef]

Cundiff, S. T.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Dekker, H.

H. Dekker, "Theory of self-locking phenomena in the pressure broadened three-mode He-Ne laser," Appl. Phys. 4, 257-263 (1974)
[CrossRef]

Demarest, F. C.

F. C. Demarest, "High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics," Meas. Sci. Technol. 9, 1024-1030 (1998).
[CrossRef]

Diddams, S. A.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Drullinger, R. E.

Evenson, K. M.

Haitjema, H.

B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, "Validation of a single fibre-fed heterodyne laser interferometer with nanometre uncertainty," Precis. Eng. 29, 229-236 (2005).
[CrossRef]

Hall, J. L.

T. H. Yoon, J. Ye, J. L. Hall, and J. -M. Chartier, "Absolute frequency measurement of the iodine-stabilized He-Ne laser at 633 nm," Appl. Phys. B 72, 221-226 (2001).

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

D. A. Jennings, C. R. Pollack, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, J. L. Hall, and H. P. Layer, "Direct frequency measurement of the I2-stabilized He-Ne 473-THz (633-nm) laser," Opt. Lett. 8, 136-138 (1983).
[CrossRef] [PubMed]

T. Baer, F. V. Kowalski, and J. L. Hall, "Frequency stabilization of a 0.633-μm He-Ne longitudinal Zeeman laser," Appl. Opt. 19, 3173-3177 (1980).
[CrossRef] [PubMed]

Hansch, T. W.

Th. Udem, R. Holzwarth, and T. W. Hansch, "Optical frequency metrology," Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Holzwarth, R.

Th. Udem, R. Holzwarth, and T. W. Hansch, "Optical frequency metrology," Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Jennings, D. A.

Jones, D. J.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Knarren, B. A. W. H.

B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, "Validation of a single fibre-fed heterodyne laser interferometer with nanometre uncertainty," Precis. Eng. 29, 229-236 (2005).
[CrossRef]

Kowalski, F. V.

Kunzmann, H.

Lamb, W. E.

W. E. Lamb, "Theory of an optical maser," Phys. Rev. 134A1429-A1450 (1964)
[CrossRef]

Lawall, J.

J. Lawall, J. M. Pedulla, and Y. L. Coq, "Ultrastable laser array at 633 nm for real-time dimensional metrology," Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

Layer, H. P.

Lebowsky, F.

Pedulla, J. M.

J. Lawall, J. M. Pedulla, and Y. L. Coq, "Ultrastable laser array at 633 nm for real-time dimensional metrology," Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

Peterson, F. R.

Pollack, C. R.

Quin, T. J.

T. J. Quin, "Mise en Pratique of the Definition of the Metre (1992)," Metrologia 30, 523-541 (1994).
[CrossRef]

Ranka, J. K.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Sayers, M. D.

M. D. Sayers and L. Allen, "Amplitude, competition, self-locking, beat frequency, and time dependent in a threemode gas laser," Phys. Rev. A 1, 1730-1746 (1970)
[CrossRef]

Schellekens, P. H. J.

B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, "Validation of a single fibre-fed heterodyne laser interferometer with nanometre uncertainty," Precis. Eng. 29, 229-236 (2005).
[CrossRef]

Suh, H. S.

H. S. Suh, T. H. Yoon, M. S. Chung, and O. S. Choi, "Frequency and power stabilization of a three longitudinal mode He-Ne laser using secondary beat frequency" Appl. Phys. Let. 63, 2027-2029 (1993)
[CrossRef]

Suzuki, N.

T. Yokoyama, T. Araki, S. Yokoyama, and N. Suzuki, "A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser" Meas. Sci. Technol. 12, 157-162 (2001)
[CrossRef]

S. Yokoyama, T. Araki, and N. Suzuki, "Intermode beat stabilized laser with frequency pulling," Appl. Opt. 33, 358-363 (1994)
[CrossRef] [PubMed]

Udem, T.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Udem, Th.

Th. Udem, R. Holzwarth, and T. W. Hansch, "Optical frequency metrology," Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

Wells, J. S.

Windeler, R. S.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Ye, J.

T. H. Yoon, J. Ye, J. L. Hall, and J. -M. Chartier, "Absolute frequency measurement of the iodine-stabilized He-Ne laser at 633 nm," Appl. Phys. B 72, 221-226 (2001).

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Yeom, J. Y.

Yokoyama, S.

T. Yokoyama, T. Araki, S. Yokoyama, and N. Suzuki, "A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser" Meas. Sci. Technol. 12, 157-162 (2001)
[CrossRef]

S. Yokoyama, T. Araki, and N. Suzuki, "Intermode beat stabilized laser with frequency pulling," Appl. Opt. 33, 358-363 (1994)
[CrossRef] [PubMed]

Yokoyama, T.

T. Yokoyama, T. Araki, S. Yokoyama, and N. Suzuki, "A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser" Meas. Sci. Technol. 12, 157-162 (2001)
[CrossRef]

Yoon, T. H.

J. Y. Yeom and T. H. Yoon, "Three-longitudinal-mode He-Ne laser frequency stabilized at 633 nm by thermal phase locking of the secondary beat frequency" Appl. Opt. 44, 266-270 (2005)
[CrossRef] [PubMed]

T. H. Yoon, J. Ye, J. L. Hall, and J. -M. Chartier, "Absolute frequency measurement of the iodine-stabilized He-Ne laser at 633 nm," Appl. Phys. B 72, 221-226 (2001).

H. S. Suh, T. H. Yoon, M. S. Chung, and O. S. Choi, "Frequency and power stabilization of a three longitudinal mode He-Ne laser using secondary beat frequency" Appl. Phys. Let. 63, 2027-2029 (1993)
[CrossRef]

Appl. Opt. (4)

Appl. Phys. (1)

H. Dekker, "Theory of self-locking phenomena in the pressure broadened three-mode He-Ne laser," Appl. Phys. 4, 257-263 (1974)
[CrossRef]

Appl. Phys. B (1)

T. H. Yoon, J. Ye, J. L. Hall, and J. -M. Chartier, "Absolute frequency measurement of the iodine-stabilized He-Ne laser at 633 nm," Appl. Phys. B 72, 221-226 (2001).

Appl. Phys. Let. (1)

H. S. Suh, T. H. Yoon, M. S. Chung, and O. S. Choi, "Frequency and power stabilization of a three longitudinal mode He-Ne laser using secondary beat frequency" Appl. Phys. Let. 63, 2027-2029 (1993)
[CrossRef]

Meas. Sci. Technol. (2)

T. Yokoyama, T. Araki, S. Yokoyama, and N. Suzuki, "A subnanometre heterodyne interferometric system with improved phase sensitivity using a three-longitudinal-mode He-Ne laser" Meas. Sci. Technol. 12, 157-162 (2001)
[CrossRef]

F. C. Demarest, "High-resolution, high-speed, low data age uncertainty, heterodyne displacement measuring interferometer electronics," Meas. Sci. Technol. 9, 1024-1030 (1998).
[CrossRef]

Metrologia (1)

T. J. Quin, "Mise en Pratique of the Definition of the Metre (1992)," Metrologia 30, 523-541 (1994).
[CrossRef]

Nature (1)

Th. Udem, R. Holzwarth, and T. W. Hansch, "Optical frequency metrology," Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. (1)

W. E. Lamb, "Theory of an optical maser," Phys. Rev. 134A1429-A1450 (1964)
[CrossRef]

Phys. Rev. A (1)

M. D. Sayers and L. Allen, "Amplitude, competition, self-locking, beat frequency, and time dependent in a threemode gas laser," Phys. Rev. A 1, 1730-1746 (1970)
[CrossRef]

Phys. Rev. Lett. (1)

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Precis. Eng. (1)

B. A. W. H. Knarren, S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, "Validation of a single fibre-fed heterodyne laser interferometer with nanometre uncertainty," Precis. Eng. 29, 229-236 (2005).
[CrossRef]

Rev. Sci. Instrum. (1)

J. Lawall, J. M. Pedulla, and Y. L. Coq, "Ultrastable laser array at 633 nm for real-time dimensional metrology," Rev. Sci. Instrum. 72, 2879-2888 (2001).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

A schematic of the three laser modes including three additional mixed modes. These modes arise from nonlinear optical interaction between V 1, V 2, and V 3.

Fig. 2.
Fig. 2.

(Color Online) Optical schematic for characterizing the three mode laser behavior. A common polarizer is used between the three different detectors and a HPF-LPF behaves like a self mixing circuit.

Fig. 3.
Fig. 3.

Measured frequency from both detectors as a function of polarization rotation. The signals are 45° out of phase, which was unexpected.

Fig. 4.
Fig. 4.

(Color Online) Optical schematic for verifying the phase between I 0 pd and Ipd 0. Two Glan-Thompson polarizers (GTP) were used to isolate the inner and outer modes, which was verified using an optical spectrum analyzer.

Fig. 5.
Fig. 5.

Comparison of I 0 pd and I 90 pd for two different three mode lasers. The mixed modal beat frequencies are 180° out of phase which shows a low speed detector cannot detect the secondary beat frequency.

Fig. 6.
Fig. 6.

Fractional frequency stability (a) and fractional noise density (b) of the center mode, μ 2, when compared with an Iodine stabilized laser using the signal generated from I 0 pd .

Equations (17)

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

v 1 + φ ˙ 1 = Ω 1 + σ 1 + ρ 1 E 1 2 + τ 12 E 2 2 + τ 13 E 3 2 ( η 23 sin Ψ ξ 23 cos Ψ ) E 2 2 E 3 E 1 1 ,
v 2 + φ ˙ 2 = Ω 2 + σ 2 + τ 21 E 1 2 + ρ 2 E 2 2 + τ 23 E 3 2 + ( η 13 sin Ψ ξ 13 cos Ψ ) E 1 E 3 , and
v 3 + φ ˙ 3 = Ω 3 + σ 3 + τ 31 E 1 2 + τ 32 E 2 2 + ρ 3 E 3 2 ( η 21 sin Ψ ξ 21 cos Ψ ) E 2 2 E 1 E 3 1 ,
Ψ = ( 2 v 2 v 1 v 3 ) t + ( 2 φ 2 φ 1 φ 3 ) = v b t + φ b
M 1 = v 1 + φ ˙ 1
M 2 = v 2 + φ ˙ 2
M 3 = v 3 + φ ˙ 3 ,
M 1 , m = v 1 + v b + φ ˙ b
M 2 , m = v 2 - v b - φ ˙ b
M 3 , m = v 3 + v b + φ ˙ b ,
I pd 0 = cos ( 2 π v b t + 3 φ 2 φ 1 φ 3 ) ,
I pd 90 = cos ( 2 πv b t + 2 φ 2 2 φ 1 φ 3 ) + cos ( 2 πv b t + 2 φ 2 φ 1 2 φ 3 ) and
I apd 90 = cos ( 2 πv 13 t + φ 3 φ 1 ) ,
I pd 90 = cos ( 2 πv b t + 2 φ 2 3 φ 13 ) and
I apd 90 = cos ( 2 π v 13 t ) .
I pd 45 = cos ( 2 π v b t + 2 φ 2 3 φ 13 ) + cos ( 2 π v b + 3 φ 2 2 φ 13 ) and
I apd 45 = cos ( 2 π v 12 t + φ 2 φ 13 ) + cos ( 2 π v 23 t + φ 13 φ 2 ) cos ( 2 π v 13 t )

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