Abstract

As laser light propagates through a resonant vapor, laser phase noise (PM) is converted to laser intensity noise (AM) because of the sensitivity of atomic coherence to laser phase fluctuations. In experiments reported here it is shown that this PM-to-AM conversion process is highly efficient and can cause the relative intensity noise of transmitted diode laser light to be 1 to 2 orders of magnitude larger than the laser’s intrinsic relative intensity noise. By use of a semiclassical description of the phenomenon, including the effect of optical pumping, reasonably good agreement between theory and experiment is obtained. The PM-to-AM conversion process discussed here has important consequences for atomic clock development, in which diode-laser optical pumping in thick alkali vapors holds the promise for orders-of-magnitude improvement in atomic clock performance.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
  2. R. Walser and P. Zoller, “Laser-noise-induced polarization fluctuations as a spectroscopic tool,” Phys. Rev. A 49, 5067 (1994).
    [CrossRef] [PubMed]
  3. D. H. McIntyre, C. E. Fairchild, J. Cooper, and R. Walser, “Diode-laser noise spectroscopy of rubidium,” Opt. Lett. 18, 1816 (1993).
    [CrossRef] [PubMed]
  4. R. J. McLean, P. Hannaford, C. E. Fairchild, and P. L. Dyson, “Tunable diode-laser heterodyne spectroscopy of atmospheric oxygen,” Opt. Lett. 18, 1675 (1993).
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  5. K. V. Vasavada, G. Vemuri, and G. S. Agarwal, “Diode-laser-noise-based spectroscopy of allowed and crossover resonances,” Phys. Rev. A 52, 4159 (1995).
    [CrossRef] [PubMed]
  6. R. Walser, J. Cooper, and P. Zoller, “Saturated absorption spectroscopy using diode-laser phase noise,” Phys. Rev. A 50, 4303 (1994).
    [CrossRef] [PubMed]
  7. J. C. Camparo and R. P. Frueholz, “A nonempirical model of the gas-cell atomic frequency standard,” J. Appl. Phys. 59, 301 (1986); “Fundamental stability limits for the diode-laser-pumped rubidium atomic frequency standard,” J. Appl. Phys. 59, 3313 (1986).
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  11. Note that (1/v)(∂E/∂t)≈(δtp/L)(E/τc), where L is the length of the medium, δtp is the propagation time of a wave front through the medium, and τc is the correlation time of the field’s stochastic variations. For a medium with a length of a few centimeters, δtp~10−10 s, whereas for the lasers of interest in our study τc~10−8 s. Thus the temporal variation of the electric field will be roughly 2 orders of magnitude smaller than the spatial variation for an optically thick medium where (∂E/∂z)≈E/L.
  12. It should be noted that in addition to Eq. (1) there is a propagation equation for the phase of the field: ∂Φ/∂z+ (1/v)(∂Φ/∂t)=−[(2πk)/n2(z, E). Inasmuch as the laser’s phase noise yields fluctuations in both χ and χ, the optical field’s phase fluctuations grow in a complicated fashion as the field propagates through the medium. The greater degree of phase noise further enhances the fluctuations of χ, which in turn results in larger variations of the laser’s transmitted intensity. However, because χ is proportional to the real part of the atomic coherence, which is generally small unless the laser has frequency excursions of the order of the optical homogeneous linewidth, it seems reasonable to ignore this additional source of laser phase noise in the PM-to-AM conversion process associated with the present experimental conditions: laser linewidth, ≅60 MHz and atomic homogeneous linewidth, ≅200 MHz. It is worth noting, though, that there will be experimental conditions in which this additional phase noise could be an important aspect of the PM-to-AM conversion process.
  13. The acronyms TJS and CSP stand for transverse junction stripe and channeled substrate planar, respectively, and refer to the physical construction of the diode laser. See D. Botez, “Single-mode AlGaAs diode lasers,” J. Opt. Commun. 1, 42 (1980) for a more detailed discussion.
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  17. P. Minguzzi and A. Di Lieto, “Simple Padé approximations for the width of a Voigt profile,” J. Mol. Spectrosc. 109, 388 (1985).
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  18. I. Botev, “A new conception of Bouguer–Lambert–Beer’s law,” Fresenius J. Anal. Chem. 297, 419 (1979).
    [CrossRef]
  19. N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, “Propagating, optical-pumping wave fronts,” Phys. Rev. Lett. 43, 519 (1979).
    [CrossRef]
  20. A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. Press, London, 1971), Chap. IV.
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  22. See J. C. Camparo and S. B. Delcamp, “Optical pumping with laser-induced-fluorescence,” Opt. Commun. 120, 257 (1995), and the discussion regarding C. H. Volk, J. C. Camparo, and R. P. Frueholz, “Investigations of laser pumped gas cell atomic frequency standard,” in Proceedings of the 13th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting (U.S. Naval Observatory, Washington, D.C., 1981), pp. 631–640.
    [CrossRef]
  23. See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
    [CrossRef] [PubMed]
  24. D. Tupa, L. W. Anderson, D. L. Huber, and J. E. Lawler, “Effect of radiation trapping on the polarization of an optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1045 (1986); D. Tupa and L. W. Anderson, “Effect of radiation trapping on the polarization of an optically pumped alkali-metal vapor in a weak magnetic field,” Phys. Rev. A 36, 2142 (1987).
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  25. Y. Yamamoto, S. Saito, and T. Mukai, “AM and FM quantum noise in semiconductor lasers. II. Comparison of theoretical and experimental results for AlGaAs lasers,” IEEE J. Quantum Electron. QE-19, 47 (1983).
    [CrossRef]
  26. H. Tsuchida and T. Tako, “Relation between frequency and intensity stabilities in AlGaAs semiconductor laser,” Jpn. J. Appl. Phys. 22, 1152 (1983).
    [CrossRef]
  27. P. Minguzzi, F. Strumia, and P. Violino, “Temperature effects in the relaxation of optically oriented alkali vapors,” Nuovo Cimento 46B, 145 (1966).
    [CrossRef]
  28. Because of the presence of the buffer gas, the atoms are essentially frozen in place. Thus individual atoms experience local values of the electric field amplitude rather than averaging the electric field amplitude over the resonance cell volume. See J. C. Camparo, R. P. Frueholz, and C. H. Volk, “Inhomogeneous light shift in alkali-metal atoms,” Phys. Rev. A 27, 1914 (1983); R. P. Frueholz and J. C. Camparo, “Microwave field strength measurement in a rubidium clock cavity via adiabatic rapid passage,” J. Appl. Phys. 57, 704 (1985).
    [CrossRef]
  29. J. C. Camparo and P. Lambropoulos, “Monte Carlo simulation of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480 (1993).
    [CrossRef] [PubMed]
  30. P. Zoller and P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
    [CrossRef]
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  32. R. H. Pennington, Introductory Computer Methods and Numerical Analysis (Macmillan, London, 1970).
  33. Note that with 10 Torr of N2 the time between velocity-changing collisions is ~10−7 s, based on a gas kinetic cross section for velocity-changing collisions, whereas the propagation time of a wave front through a 3-cm medium is ~10−10 s. Thus, on the time scale of a wave front’s propagation through the medium, each atom has a specific, essentially constant, velocity.
  34. F. A. Franz, “Relaxation at cell walls in optical pumping experiments,” Phys. Rev. A 6, 1921 (1972).
    [CrossRef]
  35. The Allan standard deviation, or Allan variance, is a statistic that is typically employed to describe precise frequency standards; it has the attractive property that it converges for certain nonstationary noise processes. Essentially, a fluctuating parameter is averaged over some time τ, and differences between neighboring averages are computed. The Allan variance is the variance associated with these differences. In the present work we take advantage of the ease with which the Allan variance may be computed and its well-known relationship to the noise process’s spectral density.
  36. J. Rutman, “Characterization of phase and frequency instabilities in precision frequency sources: fifteen years of progress,” Proc. IEEE 66, 1048 (1978); J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuningal, J. A. Mullen, Jr., W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans Instrum. Meas. IM-20, 105 (1971).
    [CrossRef]
  37. N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for Cs133, Rb87, Rb85, K39, and Na23,” Phys. Rev. 184, 102 (1969).
    [CrossRef]
  38. F. A. Franz and C. Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A 14, 1711 (1976).
    [CrossRef]
  39. T. J. Killian, “Thermionic phenomena caused by vapors of rubidium and potassium,” Phys. Rev. 27, 578 (1926).
    [CrossRef]
  40. C. Szekely, F. L. Walls, J. P. Lowe, R. E. Drullinger, and A. Novick, “Reducing local oscillator phase noise limitations on the frequency stability of passive frequency standards: tests of a new concept,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 518 (1994); Y. Saburi, Y. Koga, S. Kinugawa, T. Imamura, H. Suga, and Y. Ohuchi, “Short-term stability of laser-pumped rubidium gas cell frequency standard,” Electron. Lett. 30, 633 (1994); P. J. Chantry, I. Liberman, W. R. Verbanets, C. F. Petronio, R. L. Cather, and W. D. Partlow, “Miniature laser-pumped cesium cell atomic clock oscillator,” in Proceedings of the 1996 IEEE Frequency Control Symposium (IEEE Press, Piscataway, N.J., 1996), pp. 1002–1010; L. A. Budkin, V. L. Velichanski, A. S. Zibrov, A. A. Lyalyaskin, M. N. Penenkov, and A. I. Pikhtelev, “Double radio-optical resonance in alkali metal vapors subjected to laser excitation” Sov. J. Quantum Elecron. SJGCDU 20, 301 (1990); M. Hashimoto and M. Ohtsu, “Experiments on a semiconductor laser pumped rubidium atomic clock,” IEEE J. Quantum Electron. IEJQA7 QE-23, 446 (1987).
    [CrossRef]
  41. G. Miletti, J. Q. Deng, F. L. Walls, J. P. Lowe, and R. E. Drullinger, “Recent progress in laser-pumped rubidium gas-cell frequency standard,” in Proceedings of the 1996 IEEE International Frequency Control Symposium (IEEE Press, Piscataway, N.J., 1996), pp. 1066–1072.
  42. J. C. Camparo and W. F. Buell, “Laser PM to AM conversion in atomic vapors and short term clock stability,” in Proceedings of the 1997 IEEE International Frequency Control Symposium (IEEE Press, Piscataway, N.J., 1997), pp. 253–258.

1995 (2)

K. V. Vasavada, G. Vemuri, and G. S. Agarwal, “Diode-laser-noise-based spectroscopy of allowed and crossover resonances,” Phys. Rev. A 52, 4159 (1995).
[CrossRef] [PubMed]

See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
[CrossRef] [PubMed]

1994 (2)

R. Walser, J. Cooper, and P. Zoller, “Saturated absorption spectroscopy using diode-laser phase noise,” Phys. Rev. A 50, 4303 (1994).
[CrossRef] [PubMed]

R. Walser and P. Zoller, “Laser-noise-induced polarization fluctuations as a spectroscopic tool,” Phys. Rev. A 49, 5067 (1994).
[CrossRef] [PubMed]

1993 (3)

1991 (1)

T. Yabuzaki, T. Mitsui, and U. Tanaka, “New type of high-resolution spectroscopy with a diode laser,” Phys. Rev. Lett. 67, 2453 (1991).
[CrossRef] [PubMed]

1990 (1)

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

1985 (1)

P. Minguzzi and A. Di Lieto, “Simple Padé approximations for the width of a Voigt profile,” J. Mol. Spectrosc. 109, 388 (1985).
[CrossRef]

1983 (2)

Y. Yamamoto, S. Saito, and T. Mukai, “AM and FM quantum noise in semiconductor lasers. II. Comparison of theoretical and experimental results for AlGaAs lasers,” IEEE J. Quantum Electron. QE-19, 47 (1983).
[CrossRef]

H. Tsuchida and T. Tako, “Relation between frequency and intensity stabilities in AlGaAs semiconductor laser,” Jpn. J. Appl. Phys. 22, 1152 (1983).
[CrossRef]

1981 (1)

V. N. Belov, “Application of the magnetic-scanning method to the measurement of the broadening and shift constants of the rubidium D2 line (780.0 nm) by foreign gases,” Opt. Spectrosc. (USSR) 51, 22 (1981).

1980 (1)

The acronyms TJS and CSP stand for transverse junction stripe and channeled substrate planar, respectively, and refer to the physical construction of the diode laser. See D. Botez, “Single-mode AlGaAs diode lasers,” J. Opt. Commun. 1, 42 (1980) for a more detailed discussion.
[CrossRef]

1979 (3)

I. Botev, “A new conception of Bouguer–Lambert–Beer’s law,” Fresenius J. Anal. Chem. 297, 419 (1979).
[CrossRef]

N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, “Propagating, optical-pumping wave fronts,” Phys. Rev. Lett. 43, 519 (1979).
[CrossRef]

P. Zoller and P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

1976 (1)

F. A. Franz and C. Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A 14, 1711 (1976).
[CrossRef]

1972 (2)

F. A. Franz, “Relaxation at cell walls in optical pumping experiments,” Phys. Rev. A 6, 1921 (1972).
[CrossRef]

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169 (1972).
[CrossRef]

1969 (1)

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for Cs133, Rb87, Rb85, K39, and Na23,” Phys. Rev. 184, 102 (1969).
[CrossRef]

1966 (1)

P. Minguzzi, F. Strumia, and P. Violino, “Temperature effects in the relaxation of optically oriented alkali vapors,” Nuovo Cimento 46B, 145 (1966).
[CrossRef]

1926 (1)

T. J. Killian, “Thermionic phenomena caused by vapors of rubidium and potassium,” Phys. Rev. 27, 578 (1926).
[CrossRef]

Agarwal, G. S.

K. V. Vasavada, G. Vemuri, and G. S. Agarwal, “Diode-laser-noise-based spectroscopy of allowed and crossover resonances,” Phys. Rev. A 52, 4159 (1995).
[CrossRef] [PubMed]

Anderson, M. H.

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

Belov, V. N.

V. N. Belov, “Application of the magnetic-scanning method to the measurement of the broadening and shift constants of the rubidium D2 line (780.0 nm) by foreign gases,” Opt. Spectrosc. (USSR) 51, 22 (1981).

Bhaskar, N. D.

N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, “Propagating, optical-pumping wave fronts,” Phys. Rev. Lett. 43, 519 (1979).
[CrossRef]

Botev, I.

I. Botev, “A new conception of Bouguer–Lambert–Beer’s law,” Fresenius J. Anal. Chem. 297, 419 (1979).
[CrossRef]

Botez, D.

The acronyms TJS and CSP stand for transverse junction stripe and channeled substrate planar, respectively, and refer to the physical construction of the diode laser. See D. Botez, “Single-mode AlGaAs diode lasers,” J. Opt. Commun. 1, 42 (1980) for a more detailed discussion.
[CrossRef]

Camparo, J. C.

J. C. Camparo and P. Lambropoulos, “Monte Carlo simulation of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480 (1993).
[CrossRef] [PubMed]

Cooper, J.

R. Walser, J. Cooper, and P. Zoller, “Saturated absorption spectroscopy using diode-laser phase noise,” Phys. Rev. A 50, 4303 (1994).
[CrossRef] [PubMed]

D. H. McIntyre, C. E. Fairchild, J. Cooper, and R. Walser, “Diode-laser noise spectroscopy of rubidium,” Opt. Lett. 18, 1816 (1993).
[CrossRef] [PubMed]

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

Di Lieto, A.

P. Minguzzi and A. Di Lieto, “Simple Padé approximations for the width of a Voigt profile,” J. Mol. Spectrosc. 109, 388 (1985).
[CrossRef]

Dyson, P. L.

Elliot, D. S.

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

Fairchild, C. E.

Franz, F. A.

F. A. Franz and C. Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A 14, 1711 (1976).
[CrossRef]

F. A. Franz, “Relaxation at cell walls in optical pumping experiments,” Phys. Rev. A 6, 1921 (1972).
[CrossRef]

Hannaford, P.

Happer, W.

N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, “Propagating, optical-pumping wave fronts,” Phys. Rev. Lett. 43, 519 (1979).
[CrossRef]

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169 (1972).
[CrossRef]

Hou, M.

N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, “Propagating, optical-pumping wave fronts,” Phys. Rev. Lett. 43, 519 (1979).
[CrossRef]

Jones, R. D.

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

Killian, T. J.

T. J. Killian, “Thermionic phenomena caused by vapors of rubidium and potassium,” Phys. Rev. 27, 578 (1926).
[CrossRef]

Lambropoulos, P.

J. C. Camparo and P. Lambropoulos, “Monte Carlo simulation of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480 (1993).
[CrossRef] [PubMed]

P. Zoller and P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

Magerl, G.

See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
[CrossRef] [PubMed]

McIntyre, D. H.

McLean, R. J.

Minguzzi, P.

P. Minguzzi and A. Di Lieto, “Simple Padé approximations for the width of a Voigt profile,” J. Mol. Spectrosc. 109, 388 (1985).
[CrossRef]

P. Minguzzi, F. Strumia, and P. Violino, “Temperature effects in the relaxation of optically oriented alkali vapors,” Nuovo Cimento 46B, 145 (1966).
[CrossRef]

Mitsui, T.

T. Yabuzaki, T. Mitsui, and U. Tanaka, “New type of high-resolution spectroscopy with a diode laser,” Phys. Rev. Lett. 67, 2453 (1991).
[CrossRef] [PubMed]

Molisch, A. F.

See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
[CrossRef] [PubMed]

Mukai, T.

Y. Yamamoto, S. Saito, and T. Mukai, “AM and FM quantum noise in semiconductor lasers. II. Comparison of theoretical and experimental results for AlGaAs lasers,” IEEE J. Quantum Electron. QE-19, 47 (1983).
[CrossRef]

Oehry, B. P.

See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
[CrossRef] [PubMed]

Ressler, N. W.

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for Cs133, Rb87, Rb85, K39, and Na23,” Phys. Rev. 184, 102 (1969).
[CrossRef]

Ritsch, H.

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

Saito, S.

Y. Yamamoto, S. Saito, and T. Mukai, “AM and FM quantum noise in semiconductor lasers. II. Comparison of theoretical and experimental results for AlGaAs lasers,” IEEE J. Quantum Electron. QE-19, 47 (1983).
[CrossRef]

Sands, R. H.

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for Cs133, Rb87, Rb85, K39, and Na23,” Phys. Rev. 184, 102 (1969).
[CrossRef]

Schupita, W.

See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
[CrossRef] [PubMed]

Smith, S. J.

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

Stark, T. E.

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for Cs133, Rb87, Rb85, K39, and Na23,” Phys. Rev. 184, 102 (1969).
[CrossRef]

Strumia, F.

P. Minguzzi, F. Strumia, and P. Violino, “Temperature effects in the relaxation of optically oriented alkali vapors,” Nuovo Cimento 46B, 145 (1966).
[CrossRef]

Suleman, B.

N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, “Propagating, optical-pumping wave fronts,” Phys. Rev. Lett. 43, 519 (1979).
[CrossRef]

Sumetsberger, B.

See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
[CrossRef] [PubMed]

Tako, T.

H. Tsuchida and T. Tako, “Relation between frequency and intensity stabilities in AlGaAs semiconductor laser,” Jpn. J. Appl. Phys. 22, 1152 (1983).
[CrossRef]

Tanaka, U.

T. Yabuzaki, T. Mitsui, and U. Tanaka, “New type of high-resolution spectroscopy with a diode laser,” Phys. Rev. Lett. 67, 2453 (1991).
[CrossRef] [PubMed]

Tsuchida, H.

H. Tsuchida and T. Tako, “Relation between frequency and intensity stabilities in AlGaAs semiconductor laser,” Jpn. J. Appl. Phys. 22, 1152 (1983).
[CrossRef]

Vasavada, K. V.

K. V. Vasavada, G. Vemuri, and G. S. Agarwal, “Diode-laser-noise-based spectroscopy of allowed and crossover resonances,” Phys. Rev. A 52, 4159 (1995).
[CrossRef] [PubMed]

Vemuri, G.

K. V. Vasavada, G. Vemuri, and G. S. Agarwal, “Diode-laser-noise-based spectroscopy of allowed and crossover resonances,” Phys. Rev. A 52, 4159 (1995).
[CrossRef] [PubMed]

Violino, P.

P. Minguzzi, F. Strumia, and P. Violino, “Temperature effects in the relaxation of optically oriented alkali vapors,” Nuovo Cimento 46B, 145 (1966).
[CrossRef]

Volk, C.

F. A. Franz and C. Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A 14, 1711 (1976).
[CrossRef]

Walser, R.

R. Walser, J. Cooper, and P. Zoller, “Saturated absorption spectroscopy using diode-laser phase noise,” Phys. Rev. A 50, 4303 (1994).
[CrossRef] [PubMed]

R. Walser and P. Zoller, “Laser-noise-induced polarization fluctuations as a spectroscopic tool,” Phys. Rev. A 49, 5067 (1994).
[CrossRef] [PubMed]

D. H. McIntyre, C. E. Fairchild, J. Cooper, and R. Walser, “Diode-laser noise spectroscopy of rubidium,” Opt. Lett. 18, 1816 (1993).
[CrossRef] [PubMed]

Yabuzaki, T.

T. Yabuzaki, T. Mitsui, and U. Tanaka, “New type of high-resolution spectroscopy with a diode laser,” Phys. Rev. Lett. 67, 2453 (1991).
[CrossRef] [PubMed]

Yamamoto, Y.

Y. Yamamoto, S. Saito, and T. Mukai, “AM and FM quantum noise in semiconductor lasers. II. Comparison of theoretical and experimental results for AlGaAs lasers,” IEEE J. Quantum Electron. QE-19, 47 (1983).
[CrossRef]

Zoller, P.

R. Walser, J. Cooper, and P. Zoller, “Saturated absorption spectroscopy using diode-laser phase noise,” Phys. Rev. A 50, 4303 (1994).
[CrossRef] [PubMed]

R. Walser and P. Zoller, “Laser-noise-induced polarization fluctuations as a spectroscopic tool,” Phys. Rev. A 49, 5067 (1994).
[CrossRef] [PubMed]

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

P. Zoller and P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

Fresenius J. Anal. Chem. (1)

I. Botev, “A new conception of Bouguer–Lambert–Beer’s law,” Fresenius J. Anal. Chem. 297, 419 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Yamamoto, S. Saito, and T. Mukai, “AM and FM quantum noise in semiconductor lasers. II. Comparison of theoretical and experimental results for AlGaAs lasers,” IEEE J. Quantum Electron. QE-19, 47 (1983).
[CrossRef]

J. Mol. Spectrosc. (1)

P. Minguzzi and A. Di Lieto, “Simple Padé approximations for the width of a Voigt profile,” J. Mol. Spectrosc. 109, 388 (1985).
[CrossRef]

J. Opt. Commun. (1)

The acronyms TJS and CSP stand for transverse junction stripe and channeled substrate planar, respectively, and refer to the physical construction of the diode laser. See D. Botez, “Single-mode AlGaAs diode lasers,” J. Opt. Commun. 1, 42 (1980) for a more detailed discussion.
[CrossRef]

J. Phys. B (1)

P. Zoller and P. Lambropoulos, “Non-Lorentzian laser lineshapes in intense field–atom interaction,” J. Phys. B 12, L547 (1979).
[CrossRef]

Jpn. J. Appl. Phys. (1)

H. Tsuchida and T. Tako, “Relation between frequency and intensity stabilities in AlGaAs semiconductor laser,” Jpn. J. Appl. Phys. 22, 1152 (1983).
[CrossRef]

Nuovo Cimento (1)

P. Minguzzi, F. Strumia, and P. Violino, “Temperature effects in the relaxation of optically oriented alkali vapors,” Nuovo Cimento 46B, 145 (1966).
[CrossRef]

Opt. Lett. (2)

Opt. Spectrosc. (USSR) (1)

V. N. Belov, “Application of the magnetic-scanning method to the measurement of the broadening and shift constants of the rubidium D2 line (780.0 nm) by foreign gases,” Opt. Spectrosc. (USSR) 51, 22 (1981).

Phys. Rev. (2)

N. W. Ressler, R. H. Sands, and T. E. Stark, “Measurement of spin-exchange cross sections for Cs133, Rb87, Rb85, K39, and Na23,” Phys. Rev. 184, 102 (1969).
[CrossRef]

T. J. Killian, “Thermionic phenomena caused by vapors of rubidium and potassium,” Phys. Rev. 27, 578 (1926).
[CrossRef]

Phys. Rev. A (8)

R. Walser and P. Zoller, “Laser-noise-induced polarization fluctuations as a spectroscopic tool,” Phys. Rev. A 49, 5067 (1994).
[CrossRef] [PubMed]

F. A. Franz and C. Volk, “Spin relaxation of rubidium atoms in sudden and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A 14, 1711 (1976).
[CrossRef]

F. A. Franz, “Relaxation at cell walls in optical pumping experiments,” Phys. Rev. A 6, 1921 (1972).
[CrossRef]

J. C. Camparo and P. Lambropoulos, “Monte Carlo simulation of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480 (1993).
[CrossRef] [PubMed]

See, for example, A. F. Molisch, W. Schupita, B. P. Oehry, B. Sumetsberger, and G. Magerl, “Modeling and efficient computation of nonlinear radiation trapping in three-level atomic vapors,” Phys. Rev. A 51, 3576 (1995).
[CrossRef] [PubMed]

K. V. Vasavada, G. Vemuri, and G. S. Agarwal, “Diode-laser-noise-based spectroscopy of allowed and crossover resonances,” Phys. Rev. A 52, 4159 (1995).
[CrossRef] [PubMed]

R. Walser, J. Cooper, and P. Zoller, “Saturated absorption spectroscopy using diode-laser phase noise,” Phys. Rev. A 50, 4303 (1994).
[CrossRef] [PubMed]

See, for example, M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D. S. Elliot, H. Ritsch, and P. Zoller, “Variance and spectra of fluorescence-intensity fluctuations from two-level atoms in a phase-diffusing field,” Phys. Rev. A 42, 6690 (1990), and references therein.
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

N. D. Bhaskar, M. Hou, B. Suleman, and W. Happer, “Propagating, optical-pumping wave fronts,” Phys. Rev. Lett. 43, 519 (1979).
[CrossRef]

T. Yabuzaki, T. Mitsui, and U. Tanaka, “New type of high-resolution spectroscopy with a diode laser,” Phys. Rev. Lett. 67, 2453 (1991).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169 (1972).
[CrossRef]

Other (19)

J. C. Camparo, “The diode laser in atomic physics,” Contemp. Phys. 26, 443 (1985); C. E. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1 (1991).
[CrossRef]

A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge U. Press, London, 1971), Chap. IV.

L. Krause, “Sensitized fluorescence and quenching,” in The Excited State in Chemical Physics, J. W. McGowan, ed. (Wiley, New York, 1975), Vol. XXVIII, Chap. 4.

See J. C. Camparo and S. B. Delcamp, “Optical pumping with laser-induced-fluorescence,” Opt. Commun. 120, 257 (1995), and the discussion regarding C. H. Volk, J. C. Camparo, and R. P. Frueholz, “Investigations of laser pumped gas cell atomic frequency standard,” in Proceedings of the 13th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting (U.S. Naval Observatory, Washington, D.C., 1981), pp. 631–640.
[CrossRef]

P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, Berlin, 1991), Chap. 1.

Note that (1/v)(∂E/∂t)≈(δtp/L)(E/τc), where L is the length of the medium, δtp is the propagation time of a wave front through the medium, and τc is the correlation time of the field’s stochastic variations. For a medium with a length of a few centimeters, δtp~10−10 s, whereas for the lasers of interest in our study τc~10−8 s. Thus the temporal variation of the electric field will be roughly 2 orders of magnitude smaller than the spatial variation for an optically thick medium where (∂E/∂z)≈E/L.

It should be noted that in addition to Eq. (1) there is a propagation equation for the phase of the field: ∂Φ/∂z+ (1/v)(∂Φ/∂t)=−[(2πk)/n2(z, E). Inasmuch as the laser’s phase noise yields fluctuations in both χ and χ, the optical field’s phase fluctuations grow in a complicated fashion as the field propagates through the medium. The greater degree of phase noise further enhances the fluctuations of χ, which in turn results in larger variations of the laser’s transmitted intensity. However, because χ is proportional to the real part of the atomic coherence, which is generally small unless the laser has frequency excursions of the order of the optical homogeneous linewidth, it seems reasonable to ignore this additional source of laser phase noise in the PM-to-AM conversion process associated with the present experimental conditions: laser linewidth, ≅60 MHz and atomic homogeneous linewidth, ≅200 MHz. It is worth noting, though, that there will be experimental conditions in which this additional phase noise could be an important aspect of the PM-to-AM conversion process.

J. C. Camparo and R. P. Frueholz, “A nonempirical model of the gas-cell atomic frequency standard,” J. Appl. Phys. 59, 301 (1986); “Fundamental stability limits for the diode-laser-pumped rubidium atomic frequency standard,” J. Appl. Phys. 59, 3313 (1986).
[CrossRef]

See, for example, T. G. Vold, F. J. Raab, B. Heckel, and E. N. Fortson, “Search for a permanent electric dipole moment on the 129Xe atom,” Phys. Rev. Lett. 52, 2229 (1984); S. Appelt, G. Wackerle, M. Mehring, “A magnetic resonance study of non-adiabatic evolution of spin quantum numbers,” Z. Phys. D 34, 75 (1995).
[CrossRef]

D. Tupa, L. W. Anderson, D. L. Huber, and J. E. Lawler, “Effect of radiation trapping on the polarization of an optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1045 (1986); D. Tupa and L. W. Anderson, “Effect of radiation trapping on the polarization of an optically pumped alkali-metal vapor in a weak magnetic field,” Phys. Rev. A 36, 2142 (1987).
[CrossRef] [PubMed]

Because of the presence of the buffer gas, the atoms are essentially frozen in place. Thus individual atoms experience local values of the electric field amplitude rather than averaging the electric field amplitude over the resonance cell volume. See J. C. Camparo, R. P. Frueholz, and C. H. Volk, “Inhomogeneous light shift in alkali-metal atoms,” Phys. Rev. A 27, 1914 (1983); R. P. Frueholz and J. C. Camparo, “Microwave field strength measurement in a rubidium clock cavity via adiabatic rapid passage,” J. Appl. Phys. 57, 704 (1985).
[CrossRef]

The Allan standard deviation, or Allan variance, is a statistic that is typically employed to describe precise frequency standards; it has the attractive property that it converges for certain nonstationary noise processes. Essentially, a fluctuating parameter is averaged over some time τ, and differences between neighboring averages are computed. The Allan variance is the variance associated with these differences. In the present work we take advantage of the ease with which the Allan variance may be computed and its well-known relationship to the noise process’s spectral density.

J. Rutman, “Characterization of phase and frequency instabilities in precision frequency sources: fifteen years of progress,” Proc. IEEE 66, 1048 (1978); J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGuningal, J. A. Mullen, Jr., W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of frequency stability,” IEEE Trans Instrum. Meas. IM-20, 105 (1971).
[CrossRef]

W. Cheney and D. Kincaid, Numerical Mathematics and Computing (Brooks Cole, Monterey, Calif., 1985); W. H. Press and S. A. Teukolsky, “Adaptive stepsize Runge–Kutta integration,” Comput. Phys. 6, 188 (1992).
[CrossRef]

R. H. Pennington, Introductory Computer Methods and Numerical Analysis (Macmillan, London, 1970).

Note that with 10 Torr of N2 the time between velocity-changing collisions is ~10−7 s, based on a gas kinetic cross section for velocity-changing collisions, whereas the propagation time of a wave front through a 3-cm medium is ~10−10 s. Thus, on the time scale of a wave front’s propagation through the medium, each atom has a specific, essentially constant, velocity.

C. Szekely, F. L. Walls, J. P. Lowe, R. E. Drullinger, and A. Novick, “Reducing local oscillator phase noise limitations on the frequency stability of passive frequency standards: tests of a new concept,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 518 (1994); Y. Saburi, Y. Koga, S. Kinugawa, T. Imamura, H. Suga, and Y. Ohuchi, “Short-term stability of laser-pumped rubidium gas cell frequency standard,” Electron. Lett. 30, 633 (1994); P. J. Chantry, I. Liberman, W. R. Verbanets, C. F. Petronio, R. L. Cather, and W. D. Partlow, “Miniature laser-pumped cesium cell atomic clock oscillator,” in Proceedings of the 1996 IEEE Frequency Control Symposium (IEEE Press, Piscataway, N.J., 1996), pp. 1002–1010; L. A. Budkin, V. L. Velichanski, A. S. Zibrov, A. A. Lyalyaskin, M. N. Penenkov, and A. I. Pikhtelev, “Double radio-optical resonance in alkali metal vapors subjected to laser excitation” Sov. J. Quantum Elecron. SJGCDU 20, 301 (1990); M. Hashimoto and M. Ohtsu, “Experiments on a semiconductor laser pumped rubidium atomic clock,” IEEE J. Quantum Electron. IEJQA7 QE-23, 446 (1987).
[CrossRef]

G. Miletti, J. Q. Deng, F. L. Walls, J. P. Lowe, and R. E. Drullinger, “Recent progress in laser-pumped rubidium gas-cell frequency standard,” in Proceedings of the 1996 IEEE International Frequency Control Symposium (IEEE Press, Piscataway, N.J., 1996), pp. 1066–1072.

J. C. Camparo and W. F. Buell, “Laser PM to AM conversion in atomic vapors and short term clock stability,” in Proceedings of the 1997 IEEE International Frequency Control Symposium (IEEE Press, Piscataway, N.J., 1997), pp. 253–258.

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Figures (6)

Fig. 1
Fig. 1

(a) Experimental arrangement for the study of laser PM-to-AM conversion in an atomic medium. (b) Schematic energy-level diagram of 87Rb. In the text, on resonance refers to the 52P1/252S1/2(F=2) transition, and off resonance refers to tuning the laser midway between the two ground-state hyperfine sublevel transitions to the 52P1/2 excited state. In the theoretical model the 52S1/2(F=1), 52S1/2(F=2), and 52P1/2 states are labeled |1〉, |2〉, and |3〉, respectively.

Fig. 2
Fig. 2

Graph of the laser intensity transmitted through the resonance cell when the laser is tuned on resonance versus the incident laser intensity: filled triangles, τd=1.72 cm; open, inverted triangles, τd=1.35 cm; squares, τd=1.33 cm; circles, τd=1.31 cm.

Fig. 3
Fig. 3

Influence of radiation trapping on optical pumping: solid curves, τd=1 cm; dashed curve, τd=0.8 cm.

Fig. 4
Fig. 4

(a) RIN versus incident laser intensity: filled symbols, measurements before the laser enters the resonance cell; open symbols, measurements made after passage of the laser through the resonance cell but tuned off resonance. Triangles, τd=1.72 cm; inverted triangles, τd=1.35 cm; squares, τd=1.33 cm; circles, τd=1.31 cm. (b) RIN after passage through the resonance cell versus incident laser intensity with the laser tuned on resonance. Symbols correspond to the values of τd as in (a).

Fig. 5
Fig. 5

(a) Example of computational laser transmitted intensity variations through 3 cm of a 50 °C vapor for a laser with a 1-GHz linewidth and a RIN of 10-10. The incident laser intensity was 1 μW/cm2, and the average transmitted laser intensity was 0.22 μW/cm2. (b) Log–log plot of the Allan standard deviation σ1(τ) versus averaging time τ. The dashed line is a linear least-squares fit to the data. The slope of 0.5 for the fit indicates that the laser intensity fluctuations are white, and the intercept {log[τ(s)]=0} is a measure of the intensity fluctuations’ spectral density.

Fig. 6
Fig. 6

Computational results for the RIN of laser light transmitted through a resonant rubidium vapor: open circles, τd=1.5 cm; filled circles, τd=2.1 cm. For τd=1.5 cm the resonance cell temperature was 40 °C, and for τd=2.1 cm the resonance cell temperature was 35 °C. Error bars in the figure are semiquantitative and arise from an uncertainty in estimating the intercept {log[τ(s)]=0} of the log–log Allan standard deviation plots.

Tables (1)

Tables Icon

Table 1 Parameters Used in the Computation of the Data Presented in Fig. 6

Equations (30)

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Ez+1v Et=-2πkEn2 |χ(z, E)|,
E(z)=E0 exp-2πkn2 0z|χ(z, E)|dz.
I(z)|E(z)|2|E0|2-4πkn2 E0Pz,
σ˙1=A2 σ3-γhfs(σ1-σ2)-pA2 σ3σ1,
σ˙2=-Rσ2+A2 σ3+γhfs(σ1-σ2)-pA2 σ3σ2,
σ˙3=Rσ2-Aσ3+pA2 σ3(σ1+σ2).
σ2=-(1-p)2p+(2-p)γhfspR+(1-p)2p+(2-p)γhfspR2+(2-p)γhfspR1/2.
R(z+δz)γhfs=R(z)γhfs 1-σ2(z)δzτd.
χ(z)=Nα(z),
α(z)=2μ23E(z) Re[σ23(z)],
α(z)=2μ23E(z) Im[σ23(z)],
E(z+δz)=E(z)-8π2Nμ23n2λ |Im[σ23(z)]|vδz,
σ˙11=A2 σ33+γhfs(σ22-σ11),
σ˙22=-iΩ2 (σ32-σ23)+A2 σ33-γhfs(σ22-σ11),
σ˙33=iΩ2 (σ32-σ23)-Aσ33,
σ˙32=iΔσ32-iΩ2 (σ22-σ33)-Γc+A2σ32.
δω(t)δω(t-τ)=γβ exp(-β|τ|).
σ33(t)iΩ2A (σ32-σ23).
σ˙22=Ωv2-γhfs(2σ22-1),
u˙=-Γc+A2u-Δv,
v˙=-Γc+A2+Ω22Av+Δu-Ω2 σ22.
Σ˙=0=Ω2 v0-γhfs(2Σ-1),
u˙0=0=-Γc+A2u0-γu0-(ω0-ω32)v0,
v˙0=0=-Γc+A2+Ω22Av0-γv0+(ω0-ω32)u0-Ω2 Σ,
δ˙σ=Ω(t)2 δv-2γhfsδσ,
δ˙u=-Γc+A2δu+γu0-[ω0+δω(t)-ω32]δv-δω(t)v0,
δ˙v=-Γc+A2+Ω2(t)2Aδv+γv0+[ω0+δω(t)-ω32]δu+δω(t)u0-Ω(t)2 δσ.
x(t)x(t-τ)=γω1 exp(-ω1|τ|).
γhfs=γse+γbg+π2D4z2,0zL/2,
γhfs=γse+γbg+π2D4(L-z)2,L/2zL.

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