Abstract

We Study a modification of classical FM spectroscopy in the cases where several electromagnetic fields are FM modulated, each in a different manner. This complex spectrum scans a multi-photon resonant atomic medium with the output detected by a phase-sensitive scheme. The demodulated output signal reveals the spectroscopic features of the probed medium. The case in which two different carriers are FM modulated at the same frequency and index but with an opposite phase with respect to each other is analyzed theoretically. This configuration is essential for probing Coherent Population Trapping (CPT) resonances induced by a directly modulated diode laser. Employing a macroscopic model to describe the physical properties of CPT leads to a superb fit between predicted and measured CPT characteristics.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Bjorklund, "Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions," Opt. Lett. 5, 15-17 (1980).
    [CrossRef] [PubMed]
  2. G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
    [CrossRef]
  3. J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, "Optical heterodyne saturation spectroscopy," Appl. Phys. Lett. 39, 680-682 (1981).
    [CrossRef]
  4. E. A. Whittaker, M. Gehrtz, and G. C. Bjorklund, "Residual amplitude modulation in laser electro-optic phase modulation," J. Opt. Soc. Am. B 2, 1320-1326 (1985).
    [CrossRef]
  5. M. Gehrtz, G. Bjorklund, and E. Whittaker, "Quantum-limited laser frequency-modulation spectroscopy," J. Opt. Soc. Am. B 2, 1510-1526 (1985).
    [CrossRef]
  6. W. Lenth, "Optical heterodyne spectroscopy with frequency- and amplitude-modulated semiconductor lasers," Opt. Lett. 8, 575-577 (1983).
    [CrossRef] [PubMed]
  7. W. Lenth, "High frequency heterodyne spectroscopy with current-modulated diode lasers," IEEE J. Quantum Electron. 20, 1045-1050 (1984).
    [CrossRef]
  8. D. Cassidy and J. Reid, "Harmonic detection with tunable diode lasers - Two-tone modulation," Appl. Phys. B 29, 279-285 (1982).
    [CrossRef]
  9. G. R. Janik, C. B. Carlisle, and T. F. Gallagher, "Two-tone frequency-modulation spectroscopy," J. Opt. Soc. Am. B 3, 1070-1074 (1986).
    [CrossRef]
  10. D. E. Cooper and R. E. Warren, "Frequency modulation spectroscopy with lead-salt diode lasers: a comparison of single-tone and two-tone techniques," Appl. Opt. 26, 3726-3732 (1987).
    [CrossRef] [PubMed]
  11. C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
    [CrossRef]
  12. A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, "Influence of excited-state hyperfine structure on ground-state coherence," Phys. Rev. A 61, 012504 (1999).
    [CrossRef]
  13. E. Arimondo and G. Orriols, "Nonabsorbing Atomic Coherences by Coherent Two-Photon Transitions in a Three-Level Optical Pumping," Lett. Nouvo Cim. 17, 333-338 (1976).
    [CrossRef]
  14. E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, E. Wolf, ed., (Elsevier Science Amsterdam, 1996) Vol. 35, pp. 257-354 .
  15. A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
    [CrossRef]
  16. N. Cyr, M. T?etu, and M. Breton, "All-optical microwave frequency standard: a proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993).
    [CrossRef]
  17. J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: Dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
    [CrossRef]
  18. S. Knappe, R. Wynands, J. Kitching, H. Robinson, and L. Hollberg, "Characterization of coherent populationtrapping resonances as atomic frequency references," J. Opt. Soc. Am. B 18, 1545-1553 (2001).
    [CrossRef]
  19. J. Vanier, "Atomic clocks based on coherent population trapping: a review," Appl. Phys. B 81, 421-442 (2005).
    [CrossRef]
  20. Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, andW. Happer, "Push-Pull Optical Pumping of Pure Superposition States," Phys. Rev. Lett. 93, 160802 (2004).
    [CrossRef] [PubMed]
  21. S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, "A chip-scale atomic clock based on 87Rb with improved frequency stability," Opt. Express 13, 1249-1253 (2005).
    [CrossRef] [PubMed]
  22. R. Lutwak, P. Vlitas, M. Varghes, M. Mescher, D. K. Serkland, and G. M. Peake, "The MAC-A miniature atomic clock," in Proceedings of 2005 Joint IEEE International Frequency Control (UFFC) Symposium and the 37th Annual Precise Time & Time Interval (PTTI) Systems & Applications Meeting, D. Coler, ed., pp. 767-773 (IEEE, Vancouver, BC, Canada, 2005).
  23. I. Ben-Aroya, M. Kahanov, and G. Eisenstein, "A CPT based 87Rb atomic clock employing a small spherical glass vapor cell," in Proceedings of the 38th Annual Precise Time & Time Interval (PTTI) Systems & Applications Meeting, L. A. Breakiron, ed., pp. 259-270 (Naval Observatory, Reston, VA, USA, 2006).
  24. R. Lutwak, A. Rashed, M. Varghese, G. Tepolt, J. Leblanc, M. Mescher, D. K. Serkland, and G. M. Peake, "The Miniature Atomic Clock Pre-Production Results," in proceedings of 2005 Joint IEEE International Frequency Control (UFFC) Symposium and the 21th European Frequency and Time Forum (EFTF), D. Coler, ed., pp. 1327-1333 (IEEE, Geneva, Switzerland, 2007).
  25. M. O. Scully and M. Fleischhauer, "High-Sensitivity Magnetometer Based on Index-Enhanced Media," Phys. Rev. Lett. 69, 1360-1363 (1992).
    [CrossRef] [PubMed]
  26. P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
    [CrossRef]
  27. J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
    [CrossRef]
  28. I. Ben-Aroya, M. Kahanov, and G. Eisenstein, "Optimization of FM spectroscopy parameters for a frequency locking loop in small scale CPT based atomic clocks," Opt. Express 15, 15060-15065 (2007).
    [CrossRef] [PubMed]
  29. J. A. Silver, "Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods," Appl. Opt. 31, 707-717 (1992).
    [CrossRef] [PubMed]
  30. J. M. Supplee, E. A. Whittaker, and W. Lenth, "Theoretical description of frequency modulation and wavelength modulation spectroscopy," Appl. Opt. 33, 6294-6302 (1994).
    [CrossRef] [PubMed]
  31. R. Wynands and A. Nagel, "Inversion of frequency-modulation spectroscopy line shapes," J. Opt. Soc. Am. B 16, 1617-1622 (1999).
    [CrossRef]
  32. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, New York, 1997). Ch. 5.
  33. M. Kahanov, Electrical Engineering department, Technion, Haifa 32000, Israel. (personal communication, 2007).
  34. C. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
    [CrossRef]
  35. X. Zhu and D. T. Cassidy, "Modulation spectroscopy with a semiconductor diode laser by injection-current modulation," J. Opt. Soc. Am. B 14, 1945-1950 (1997).
    [CrossRef]
  36. A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, "Anomalous Interaction of Spectral Modes in a Semiconductor Laser," IEEE J. Quantum Electron. 11, 510-515 (1975).
    [CrossRef]
  37. D. Phillips, I. Novikova, C. Wang, R. Walsworth, and M. Crescimanno, "Modulation-induced frequency shifts in a coherent-population-trapping-based atomic clock," J. Opt. Soc. Am. B 22, 305-310 (2005).
    [CrossRef]

2007

2005

2004

Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, andW. Happer, "Push-Pull Optical Pumping of Pure Superposition States," Phys. Rev. Lett. 93, 160802 (2004).
[CrossRef] [PubMed]

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

2003

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

2001

2000

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

1999

A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, "Influence of excited-state hyperfine structure on ground-state coherence," Phys. Rev. A 61, 012504 (1999).
[CrossRef]

R. Wynands and A. Nagel, "Inversion of frequency-modulation spectroscopy line shapes," J. Opt. Soc. Am. B 16, 1617-1622 (1999).
[CrossRef]

1998

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: Dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

1997

1994

1993

N. Cyr, M. T?etu, and M. Breton, "All-optical microwave frequency standard: a proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993).
[CrossRef]

1992

J. A. Silver, "Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods," Appl. Opt. 31, 707-717 (1992).
[CrossRef] [PubMed]

M. O. Scully and M. Fleischhauer, "High-Sensitivity Magnetometer Based on Index-Enhanced Media," Phys. Rev. Lett. 69, 1360-1363 (1992).
[CrossRef] [PubMed]

1987

1986

1985

1984

W. Lenth, "High frequency heterodyne spectroscopy with current-modulated diode lasers," IEEE J. Quantum Electron. 20, 1045-1050 (1984).
[CrossRef]

1983

W. Lenth, "Optical heterodyne spectroscopy with frequency- and amplitude-modulated semiconductor lasers," Opt. Lett. 8, 575-577 (1983).
[CrossRef] [PubMed]

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
[CrossRef]

1982

D. Cassidy and J. Reid, "Harmonic detection with tunable diode lasers - Two-tone modulation," Appl. Phys. B 29, 279-285 (1982).
[CrossRef]

C. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
[CrossRef]

1981

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, "Optical heterodyne saturation spectroscopy," Appl. Phys. Lett. 39, 680-682 (1981).
[CrossRef]

1980

1976

E. Arimondo and G. Orriols, "Nonabsorbing Atomic Coherences by Coherent Two-Photon Transitions in a Three-Level Optical Pumping," Lett. Nouvo Cim. 17, 333-338 (1976).
[CrossRef]

1975

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, "Anomalous Interaction of Spectral Modes in a Semiconductor Laser," IEEE J. Quantum Electron. 11, 510-515 (1975).
[CrossRef]

Affolderbach, C.

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, "Influence of excited-state hyperfine structure on ground-state coherence," Phys. Rev. A 61, 012504 (1999).
[CrossRef]

Arimondo, E.

E. Arimondo and G. Orriols, "Nonabsorbing Atomic Coherences by Coherent Two-Photon Transitions in a Three-Level Optical Pumping," Lett. Nouvo Cim. 17, 333-338 (1976).
[CrossRef]

Baer, T.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, "Optical heterodyne saturation spectroscopy," Appl. Phys. Lett. 39, 680-682 (1981).
[CrossRef]

Ben-Aroya, I.

Bjorklund, G.

Bjorklund, G. C.

Bogatov, A. P.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, "Anomalous Interaction of Spectral Modes in a Semiconductor Laser," IEEE J. Quantum Electron. 11, 510-515 (1975).
[CrossRef]

Carlisle, C. B.

Cassidy, D.

D. Cassidy and J. Reid, "Harmonic detection with tunable diode lasers - Two-tone modulation," Appl. Phys. B 29, 279-285 (1982).
[CrossRef]

Cassidy, D. T.

Cooper, D. E.

Crescimanno, M.

Cyr, N.

N. Cyr, M. T?etu, and M. Breton, "All-optical microwave frequency standard: a proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993).
[CrossRef]

Eisenstein, G.

Eliseev, P. G.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, "Anomalous Interaction of Spectral Modes in a Semiconductor Laser," IEEE J. Quantum Electron. 11, 510-515 (1975).
[CrossRef]

Fleischhauer, M.

M. O. Scully and M. Fleischhauer, "High-Sensitivity Magnetometer Based on Index-Enhanced Media," Phys. Rev. Lett. 69, 1360-1363 (1992).
[CrossRef] [PubMed]

Gallagher, T. F.

Gehrtz, M.

Godone, A.

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: Dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

Hall, J. L.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, "Optical heterodyne saturation spectroscopy," Appl. Phys. Lett. 39, 680-682 (1981).
[CrossRef]

Henry, C.

C. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
[CrossRef]

Hollberg, L.

S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, "A chip-scale atomic clock based on 87Rb with improved frequency stability," Opt. Express 13, 1249-1253 (2005).
[CrossRef] [PubMed]

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

S. Knappe, R. Wynands, J. Kitching, H. Robinson, and L. Hollberg, "Characterization of coherent populationtrapping resonances as atomic frequency references," J. Opt. Soc. Am. B 18, 1545-1553 (2001).
[CrossRef]

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, "Optical heterodyne saturation spectroscopy," Appl. Phys. Lett. 39, 680-682 (1981).
[CrossRef]

Janik, G. R.

Jau, Y.-Y.

Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, andW. Happer, "Push-Pull Optical Pumping of Pure Superposition States," Phys. Rev. Lett. 93, 160802 (2004).
[CrossRef] [PubMed]

Jung, C.

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

Kahanov, M.

Kitching, J.

S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, "A chip-scale atomic clock based on 87Rb with improved frequency stability," Opt. Express 13, 1249-1253 (2005).
[CrossRef] [PubMed]

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

S. Knappe, R. Wynands, J. Kitching, H. Robinson, and L. Hollberg, "Characterization of coherent populationtrapping resonances as atomic frequency references," J. Opt. Soc. Am. B 18, 1545-1553 (2001).
[CrossRef]

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

Knappe, S.

S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, "A chip-scale atomic clock based on 87Rb with improved frequency stability," Opt. Express 13, 1249-1253 (2005).
[CrossRef] [PubMed]

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

S. Knappe, R. Wynands, J. Kitching, H. Robinson, and L. Hollberg, "Characterization of coherent populationtrapping resonances as atomic frequency references," J. Opt. Soc. Am. B 18, 1545-1553 (2001).
[CrossRef]

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, "Influence of excited-state hyperfine structure on ground-state coherence," Phys. Rev. A 61, 012504 (1999).
[CrossRef]

Kuzma, N. N.

Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, andW. Happer, "Push-Pull Optical Pumping of Pure Superposition States," Phys. Rev. Lett. 93, 160802 (2004).
[CrossRef] [PubMed]

Lenth, W.

J. M. Supplee, E. A. Whittaker, and W. Lenth, "Theoretical description of frequency modulation and wavelength modulation spectroscopy," Appl. Opt. 33, 6294-6302 (1994).
[CrossRef] [PubMed]

W. Lenth, "High frequency heterodyne spectroscopy with current-modulated diode lasers," IEEE J. Quantum Electron. 20, 1045-1050 (1984).
[CrossRef]

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
[CrossRef]

W. Lenth, "Optical heterodyne spectroscopy with frequency- and amplitude-modulated semiconductor lasers," Opt. Lett. 8, 575-577 (1983).
[CrossRef] [PubMed]

Levenson, M.

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
[CrossRef]

Levi, F.

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: Dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

Liew, L.

Liew, L.-A.

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

Miron, E.

Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, andW. Happer, "Push-Pull Optical Pumping of Pure Superposition States," Phys. Rev. Lett. 93, 160802 (2004).
[CrossRef] [PubMed]

Moreland, J.

S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, "A chip-scale atomic clock based on 87Rb with improved frequency stability," Opt. Express 13, 1249-1253 (2005).
[CrossRef] [PubMed]

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

Nagel, A.

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, "Influence of excited-state hyperfine structure on ground-state coherence," Phys. Rev. A 61, 012504 (1999).
[CrossRef]

R. Wynands and A. Nagel, "Inversion of frequency-modulation spectroscopy line shapes," J. Opt. Soc. Am. B 16, 1617-1622 (1999).
[CrossRef]

Novikova, I.

Orriols, G.

E. Arimondo and G. Orriols, "Nonabsorbing Atomic Coherences by Coherent Two-Photon Transitions in a Three-Level Optical Pumping," Lett. Nouvo Cim. 17, 333-338 (1976).
[CrossRef]

Ortiz, C.

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
[CrossRef]

Phillips, D.

Post, A. B.

Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, andW. Happer, "Push-Pull Optical Pumping of Pure Superposition States," Phys. Rev. Lett. 93, 160802 (2004).
[CrossRef] [PubMed]

Reid, J.

D. Cassidy and J. Reid, "Harmonic detection with tunable diode lasers - Two-tone modulation," Appl. Phys. B 29, 279-285 (1982).
[CrossRef]

Robinson, H.

Robinson, H. G.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, "Optical heterodyne saturation spectroscopy," Appl. Phys. Lett. 39, 680-682 (1981).
[CrossRef]

Schwindt, P.

Schwindt, P. D. D.

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

Scully, M. O.

M. O. Scully and M. Fleischhauer, "High-Sensitivity Magnetometer Based on Index-Enhanced Media," Phys. Rev. Lett. 69, 1360-1363 (1992).
[CrossRef] [PubMed]

Shah, V.

S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, "A chip-scale atomic clock based on 87Rb with improved frequency stability," Opt. Express 13, 1249-1253 (2005).
[CrossRef] [PubMed]

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

Silver, J. A.

Stahler, M.

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

Supplee, J. M.

Sverdlov, B. N.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, "Anomalous Interaction of Spectral Modes in a Semiconductor Laser," IEEE J. Quantum Electron. 11, 510-515 (1975).
[CrossRef]

Taichenachev, A.

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

Vanier, J.

J. Vanier, "Atomic clocks based on coherent population trapping: a review," Appl. Phys. B 81, 421-442 (2005).
[CrossRef]

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: Dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

Vukicevic, M.

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

Walsworth, R.

Wang, C.

Warren, R. E.

Weidmann, W.

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

Whittaker, E.

Whittaker, E. A.

Wiedenmann, D.

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

Wynands, R.

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

S. Knappe, R. Wynands, J. Kitching, H. Robinson, and L. Hollberg, "Characterization of coherent populationtrapping resonances as atomic frequency references," J. Opt. Soc. Am. B 18, 1545-1553 (2001).
[CrossRef]

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

R. Wynands and A. Nagel, "Inversion of frequency-modulation spectroscopy line shapes," J. Opt. Soc. Am. B 16, 1617-1622 (1999).
[CrossRef]

A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, "Influence of excited-state hyperfine structure on ground-state coherence," Phys. Rev. A 61, 012504 (1999).
[CrossRef]

Yudin, V.

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

Zhu, X.

Appl. Opt.

Appl. Phys. B

G. Bjorklund, M. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
[CrossRef]

D. Cassidy and J. Reid, "Harmonic detection with tunable diode lasers - Two-tone modulation," Appl. Phys. B 29, 279-285 (1982).
[CrossRef]

C. Affolderbach, A. Nagel, S. Knappe, C. Jung, D. Wiedenmann, and R. Wynands, "Nonlinear spectroscopy with a vertical-cavity surface-emitting laser (VCSEL)," Appl. Phys. B 70, 407-413 (2000).
[CrossRef]

J. Vanier, "Atomic clocks based on coherent population trapping: a review," Appl. Phys. B 81, 421-442 (2005).
[CrossRef]

Appl. Phys. Lett.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, "Optical heterodyne saturation spectroscopy," Appl. Phys. Lett. 39, 680-682 (1981).
[CrossRef]

P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg, J. Kitching, L.-A. Liew, and J. Moreland, "Chip-scale atomic magnetometer," Appl. Phys. Lett. 85, 6409-6411 (2004).
[CrossRef]

IEEE J. Quantum Electron.

C. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
[CrossRef]

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, "Anomalous Interaction of Spectral Modes in a Semiconductor Laser," IEEE J. Quantum Electron. 11, 510-515 (1975).
[CrossRef]

W. Lenth, "High frequency heterodyne spectroscopy with current-modulated diode lasers," IEEE J. Quantum Electron. 20, 1045-1050 (1984).
[CrossRef]

IEEE Trans. Instrum. Meas.

N. Cyr, M. T?etu, and M. Breton, "All-optical microwave frequency standard: a proposal," IEEE Trans. Instrum. Meas. 42, 640-649 (1993).
[CrossRef]

J. Kitching, S. Knappe, M. Vukicevic, L. Hollberg, R. Wynands, and W. Weidmann, "A microwave frequency reference based on VCSEL-driven dark lineresonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Lett. Nouvo Cim.

E. Arimondo and G. Orriols, "Nonabsorbing Atomic Coherences by Coherent Two-Photon Transitions in a Three-Level Optical Pumping," Lett. Nouvo Cim. 17, 333-338 (1976).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

A. Taichenachev, V. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 33810 (2003).
[CrossRef]

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: Dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

A. Nagel, C. Affolderbach, S. Knappe, and R. Wynands, "Influence of excited-state hyperfine structure on ground-state coherence," Phys. Rev. A 61, 012504 (1999).
[CrossRef]

Phys. Rev. Lett.

M. O. Scully and M. Fleischhauer, "High-Sensitivity Magnetometer Based on Index-Enhanced Media," Phys. Rev. Lett. 69, 1360-1363 (1992).
[CrossRef] [PubMed]

Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, andW. Happer, "Push-Pull Optical Pumping of Pure Superposition States," Phys. Rev. Lett. 93, 160802 (2004).
[CrossRef] [PubMed]

Other

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University Press, New York, 1997). Ch. 5.

M. Kahanov, Electrical Engineering department, Technion, Haifa 32000, Israel. (personal communication, 2007).

E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, E. Wolf, ed., (Elsevier Science Amsterdam, 1996) Vol. 35, pp. 257-354 .

R. Lutwak, P. Vlitas, M. Varghes, M. Mescher, D. K. Serkland, and G. M. Peake, "The MAC-A miniature atomic clock," in Proceedings of 2005 Joint IEEE International Frequency Control (UFFC) Symposium and the 37th Annual Precise Time & Time Interval (PTTI) Systems & Applications Meeting, D. Coler, ed., pp. 767-773 (IEEE, Vancouver, BC, Canada, 2005).

I. Ben-Aroya, M. Kahanov, and G. Eisenstein, "A CPT based 87Rb atomic clock employing a small spherical glass vapor cell," in Proceedings of the 38th Annual Precise Time & Time Interval (PTTI) Systems & Applications Meeting, L. A. Breakiron, ed., pp. 259-270 (Naval Observatory, Reston, VA, USA, 2006).

R. Lutwak, A. Rashed, M. Varghese, G. Tepolt, J. Leblanc, M. Mescher, D. K. Serkland, and G. M. Peake, "The Miniature Atomic Clock Pre-Production Results," in proceedings of 2005 Joint IEEE International Frequency Control (UFFC) Symposium and the 21th European Frequency and Time Forum (EFTF), D. Coler, ed., pp. 1327-1333 (IEEE, Geneva, Switzerland, 2007).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

a) Optical spectrum of conventional FM spectroscopy. ; b) Optical spectrum of DFFMS. The carrier frequencies are ω c1,2=ωo ±ωµ and ωm is the FM modulation frequency. The FM modulated carriers are shown with the side bands up to second order. The green arrows represent tuning directions for an increased RF frequency (ωµ ) as the respective spectra scan a two-photon resonance. ; c) The atomic energy structure of the 87 Rb D 2 transition three-level Λ-system.

Fig. 2.
Fig. 2.

A comparison between the conventional FM spectroscopy and the DFFMS (simulated up to the first order). A Lorentzian medium with Γ=1 Hz was used in both cases The FM parameters are: a) fm =1 Hz, M=0.5 ; b) fm =1 Hz, M=1 ; c) fm =2 Hz, M=1. The horizontal axis describes the detuning between the two carriers relative to the resonance frequency: Δf=(ω c1-ω c2)-ωres . The amplitude in each method was normalized separately.

Fig. 3.
Fig. 3.

The experimental setup (ND-Natural Density filter, Lin Pol-Linear Polarizer, λ/4-Quarter wave plate).

Fig. 4.
Fig. 4.

A comparison between the Double-Field FM Spectroscopy experimental and simulation results. The figure presents the amplitude of the two Lock-In amplifier output components, namely X and Y, versus frequency detuning from resonance for four different FM parameter sets, as appear in the bottom-left corner of each graph. The residual errors of the fittings are presented under each graph (simulation-measurement).

Fig. 5.
Fig. 5.

The effect of the RAM on the DFFMS. All figures present output components versus frequency detuning shown in polar representation. The simulation results presented in figures (a), (b) and (c) do not include RAM terms (i.e. R 1=R 2=0). a) R and θ components of the measured and simulated output. ; b) The θ component (simulated and measured) of (a) after neglecting the π jumps between quarters. ; c) An enlargements of (b) around the center. Figures (d), (e) and (f) include RAM terms R 1=1E-3,R 2=0.99E-3,ψ1 =0,ψ2 =0.997π rad. d) R and θ components of the measured and simulated output. ; e) The θ component (simulated and measured) of (d) after neglecting the π jumps between quarters.; f) An enlargements of (e) around the center.

Equations (68)

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

E ( t ) = A cos ( ω c t ) = 1 2 [ A e i ω c t + c . c . ] = 1 2 [ E ̃ ( t ) + c . c . ]
E ( t ) = A cos ( ω c t + M · sin ( ω m t ) ) =
= 1 2 [ A n = J n ( M ) e i ( ( ω c + n ω m ) t ) + c . c . ]
T ω = T ¯ · exp { δ ( ω ) i ϕ ( ω ) }
E ( t ) = 1 2 [ A n = J n ( M ) e i ( ( ω c + n ω m ) t ) T ( ω c + n ω m ) + c . c . ]
i-p : A 2 J 0 ( M ) J 1 ( M ) [ e ( δ 1 + δ 0 ) cos ( ϕ 1 ϕ 0 ) e ( δ 0 + δ 1 ) cos ( ϕ 0 ϕ 1 ) ] quad : A 2 J 0 ( M ) J 1 ( M ) [ e ( δ 1 + δ 0 ) sin ( ϕ 1 ϕ 0 ) e ( δ 0 + δ 1 ) sin ( ϕ 0 ϕ 1 ) ]
i-p : A 2 n = 1 N J n ( M ) J n 1 ( M ) e ( δ n + δ n 1 ) cos ( ϕ n ϕ n 1 ) . . .
J ( n 1 ) ( M ) J n ( M ) e ( δ ( n 1 ) + δ n ) cos ( ϕ ( n 1 ) ϕ n )
quad : A 2 n = 1 N J n ( M ) J n 1 ( M ) e ( δ n + δ n 1 ) sin ( ϕ n ϕ n 1 ) . . .
J ( n 1 ) ( M ) J n ( M ) e ( δ ( n 1 ) + δ n ) sin ( ϕ ( n 1 ) ϕ n )
E ( t ) = E 1 ( t ) + E 2 ( t ) = 1 2 [ E ~ ( t ) + c . c . ]
E 1 ( t ) = A 1 cos ( ω c 1 t + M · sin ( ω m t ) ) = . . .
= 1 2 [ A 1 n = N 1 N 1 J n ( M ) e i ( ( ω c 1 + n ω m ) t ) + c . c . ] = 1 2 [ E ~ 1 ( t ) + c . c . ]
E 2 ( t ) = A 2 cos ( ω c 2 t M · sin ( ω m t ) ) = . . .
= 1 2 [ A 2 n = N 2 N 2 J n ( M ) e i ( ( ω c 2 + n ω m ) t ) + c . c . ] = . . .
= 1 2 [ A 2 n = N 2 N 2 J n ( M ) e i ( ( ω c 2 n ω m ) t ) + c . c . ] = 1 2 [ E ~ 2 ( t ) + c . c . ]
E ~ ( t ) = A 1 n 1 = N 1 N 1 J n 1 ( M ) e i ( ( ω c 1 + n 1 ω m ) t ) · n 2 = N 2 N 2 w n 2 ( 2 ) T ( 2 ω μ + ( n 1 n 2 ) ω m ) ( 1 ) + . . .
+ A 2 n 2 = N 2 N 2 J n 2 ( M ) e i ( ( ω c 2 + n 2 ω m ) t ) · n 1 = N 1 N 1 w n 1 ( 1 ) T ( 2 ω μ + ( n 1 n 2 ) ω m ) ( 2 )
T ( ω ) ( i ) = T ¯ · e ( δ ( ω ) ( i ) i ϕ ( ω ) ( i ) ) ; i = { 1 , 2 } ; ω c 1 > ω c 2
δ ( ω ) ( 1 ) = δ ( ω ) ( 2 ) = a · ( 1 R ( ω ) 2 + 1 )
ϕ ( ω ) ( 1 ) = ϕ ( ω ) ( 2 ) = a · ( R ( ω ) R ( ω ) 2 + 1 )
R ( ω ) = ω ω res Γ 2
w n ( 2 ) = A 2 2 J n ( M ) 2
w n ( 1 ) = A 1 2 J n ( M ) 2
E ~ ( t ) = E ~ 1 T + E ~ 2 T
E ~ 1 T = A 1 A 2 2 2 {
J 1 ( M ) e i ( ( ω c 1 ω m ) t ) [ J 1 ( M ) 2 T ( 2 ω μ ) ( 1 ) + J 0 ( M ) 2 T ( 2 ω μ ω m ) ( 1 ) + J 1 ( M ) 2 T ( 2 ω μ 2 ω m ) ( 1 ) ] + . . .
+ J 0 ( M ) e i ( ω c 1 t ) [ J 1 ( M ) 2 T ( 2 ω μ + ω m ) ( 1 ) + J 0 ( M ) 2 T ( 2 ω μ ) ( 1 ) + J 1 ( M ) 2 T ( 2 ω μ ω m ) ( 1 ) ] + . . .
+ J 1 ( M ) e i ( ( ω c 1 ω m ) t ) [ J 1 ( M ) 2 T ( 2 ω μ + 2 ω m ) ( 1 ) + J 0 ( M ) 2 T ( 2 ω μ ω m ) ( 1 ) + J 1 ( M ) 2 T ( 2 ω μ ) ( 1 ) ] }
E ~ 2 T = A 2 A 1 2 2 {
J 1 ( M ) e i ( ( ω c 2 ω m ) t ) [ J 1 ( M ) 2 T ( 2 ω μ ) ( 2 ) + J 0 ( M ) 2 T ( 2 ω μ + ω m ) ( 2 ) + J 1 ( M ) 2 T ( 2 ω μ 2 ω m ) ( 2 ) ] + . . .
+ J 0 ( M ) e i ( ω c 2 t ) [ J 1 ( M ) 2 T ( 2 ω μ ω m ) ( 2 ) + J 0 ( M ) 2 T ( 2 ω μ ) ( 2 ) + J 1 ( M ) 2 T ( 2 ω μ ω m ) ( 2 ) ] + . . .
+ J 1 ( M ) e i ( ( ω c 2 ω m ) t ) [ J 1 ( M ) 2 T ( 2 ω μ 2 ω m ) ( 2 ) + J 0 ( M ) 2 T ( 2 ω μ ω m ) ( 2 ) + J 1 ( M ) 2 T ( 2 ω μ ) ( 2 ) ] }
I E ~ 2 = E ~ 1 T + E ~ 2 T 2 = E ~ 1 T 2 + E ~ 2 T 2 + E ~ 1 T E ~ 2 T * + E ~ 2 T E ~ 1 T *
E ~ 1 T 2 i-p : η 1 { J 0 ( M ) 4 C 10 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 C 20 ( 1 ) + J 1 ( M ) 4 C 21 ( 1 ) + J 1 ( M ) 4 C 2 1 ( 1 ) + . . .
+ J 0 ( M ) 2 J 1 ( M ) 2 [ e 2 δ 1 ( 1 ) e 2 δ 1 ( 1 ) ] }
E ~ 1 T 2 quad : η 1 { J 0 ( M ) 4 S 10 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 S 20 ( 1 ) + J 1 ( M ) 4 S 21 ( 1 ) + J 1 ( M ) 4 S 2 1 ( 1 ) . . .
2 J 1 ( M ) 4 S 10 ( 1 ) }
E ~ 2 T 2 i-p : η 2 { J 0 ( M ) 4 C 10 ( 2 ) J 0 ( M ) 2 J 1 ( M ) 2 C 20 ( 2 ) J 1 ( M ) 4 C 21 ( 2 ) J 1 ( M ) 4 C 2 1 ( 2 ) . . .
J 0 ( M ) 2 J 1 ( M ) 2 [ e 2 δ 1 ( 2 ) e 2 δ 1 ( 2 ) ] }
E ~ 2 T 2 quad : η 2 { J 0 ( M ) 4 S 10 ( 2 ) + J 0 ( M ) 2 J 1 ( M ) 2 S 20 ( 2 ) + J 1 ( M ) 4 S 21 ( 1 ) + J 1 ( M ) 4 S 2 1 ( 2 ) . . .
2 J 1 ( M ) 4 S 10 ( 2 ) }
η 1 = 1 2 J 0 ( M ) J 1 ( M ) A 1 A 2 2 2 T ¯ 2 ; η 2 = 1 2 J 0 ( M ) J 1 ( M ) A 1 2 A 2 2 T ¯ 2
C nl ( i ) = e ( δ n ( i ) + δ l ( i ) ) cos ( ϕ n ( i ) ϕ l ( i ) ) e ( δ l ( i ) + δ n ( i ) ) cos ( ϕ l ( i ) ϕ n ( i ) )
S nl ( i ) = e ( δ n ( i ) + δ l ( i ) ) sin ( ϕ n ( i ) ϕ l ( i ) ) e ( δ l ( i ) + δ n ( i ) ) sin ( ϕ l ( i ) ϕ n ( i ) )
δ n ( i ) = δ ( 2 ω μ + n ω m ) ( i ) ; ϕ n ( i ) = ϕ ( 2 ω μ + n ω m ) ( i ) ; i = { 1,2 }
E ~ 1 T 2 i-p : η 1 { J 0 ( M ) 4 C 10 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 C 20 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 [ e 2 δ 1 ( 1 ) e 2 δ 1 ( 1 ) ] }
E ~ 1 T 2 quad : η 1 { J 0 ( M ) 4 S 10 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 S 20 ( 1 ) }
E ~ 2 T 2 i-p : η 2 { J 0 ( M ) 4 C 10 ( 2 ) + J 0 ( M ) 2 J 1 ( M ) 2 C 20 ( 2 ) + J 0 ( M ) 2 J 1 ( M ) 2 [ e 2 δ 1 ( 2 ) e 2 δ 1 ( 2 ) ] }
E ~ 2 T 2 quad : η 2 { J 0 ( M ) 4 S 10 ( 2 ) + J 0 ( M ) 2 J 1 ( M ) 2 S 20 ( 2 ) }
i-p : 2 η 1 { J 0 ( M ) 4 C 10 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 C 20 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 [ e 2 δ 1 ( 1 ) e 2 δ 1 ( 1 ) ] }
quad : 2 η 1 { J 0 ( M ) 4 S 10 ( 1 ) + J 0 ( M ) 2 J 1 ( M ) 2 S 20 ( 1 ) }
i-p : A 2 J 0 ( M ) J 1 ( M ) C 10
quad : A 2 J 0 ( M ) J 1 ( M ) S 10
E 1 ( t ) = A 1 cos ( ω c 1 t + M . sin ( ω m t ) ) [ 1 + R 1 sin ( ω m t + ψ 1 ) ] 1 2
E ~ 1 ( t ) = A 1 n = N 1 N 1 J n ( M ) e i ( ( ω c + n ω m ) t ) [ 1 i R 1 2 e i ( ω m t + ψ 1 ) + i R 1 2 e i ( ω m t + ψ 1 ) ] 1 2
E 2 ( t ) = A 2 cos ( ω c 2 t - M . sin ( ω m t ) ) [ 1 + R 2 sin ( ω m t + ψ 2 ) ] 1 2
E ~ 2 ( t ) = A 2 n = N 2 N 2 J n ( M ) e i ( ( ω c + n ω m ) t ) [ 1 i R 2 2 e i ( ω m t + ψ 2 ) + i R 2 2 e i ( ω m t + ψ 2 ) ] 1 2
E 1 ( t ) A 1 cos ( ω c 1 t + M · sin ( ω m t ) ) [ 1 + R 1 2 sin ( ω m t + ψ 1 ) ]
E ~ 1 ( t ) A 1 n = N 1 N 1 J n ( M ) e i ( ( ω c + n ω m ) t ) [ 1 i R 1 4 e i ( ω m t + ψ 1 ) + i R 1 4 e i ( ω m t + ψ 1 ) ]
E 2 ( t ) A 2 cos ( ω c 2 t + M · sin ( ω m t ) ) [ 1 + R 2 2 sin ( ω m t + ψ 2 ) ]
E ~ 2 ( t ) A 2 n = N 2 N 2 J n ( M ) e i ( ( ω c + n ω m ) t ) [ 1 i R 2 4 e i ( ω m t + ψ 2 ) + i R 2 4 e i ( ω m t + ψ 2 ) ]
E ~ ( t ) = A 1 n 1 = N 1 N 1 ( J n 1 ( M ) i J n 1 1 ( M ) R 1 4 e i ψ 1 + i J n 1 + 1 ( M ) R 1 4 e i ψ 1 ) e i ( ( ω c 1 + n 1 ω m ) t ) .
. n 2 = N 2 N 2 w m ( 2 ) T ( 2 ω m + ( n 1 n 2 ) ω m ) ( 1 ) +
+ A 2 n 2 = N 2 N 2 ( J n 2 ( M ) i J n 2 1 ( M ) R 2 4 e i ψ 2 + i J n 2 + 1 ( M ) R 2 4 e i ψ 2 ) e i ( ( ω c 2 + n 2 ω m ) t ) .
. n 1 = N 1 N 1 w n 1 ( 1 ) T ( 2 ω m + ( n 1 n 2 ) ω m ) ( 1 )
w n ( 2 ) = A 2 2 ( J n ( M ) i J n 1 ( M ) R 2 4 e i ψ 2 + i J n + 1 ( M ) R 2 4 e i ψ 2 ) 2
w n ( 1 ) = A 1 2 ( J n ( M ) i J n 1 ( M ) R 1 4 e i ψ 1 + i J n + 1 ( M ) R 1 4 e i ψ 1 ) 2

Metrics