Abstract

We present a comprehensive and quasi-tutorial review of the theory for analyzing the optical power spectrum of an optical field that has noise modulations of both the amplitude and the phase. We also present experimental results of the frequency stabilization of a commercial dye laser to a high-finesse Fabry–Perot cavity (0.49-Hz resulting full linewidth) and of the optical phase locking of the dye laser to a second reference laser (putting 97% of the optical power into the carrier) using an external stabilizer scheme. This external optical phase/frequency stabilization technique can be applied to virtually any cw laser system.

© 1993 Optical Society of America

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  1. J. L. Jespersen, B. E. Blair, and L. E. Gatterer, eds., special issue on time and frequency, Proc. IEEE 60, 476 ff. (1972).
    [CrossRef]
  2. J. L. Jespersen and D. W. Hanson eds., special issue on time and frequency, Proc. IEEE 79, 894 ff. (1991).
    [CrossRef]
  3. Comité International des Poids et Mesures, Recommendation 1 (CI-1983). See “Editor’s note,” Metrologia 19, 163 (1984).
  4. Comité International des Poids et Mesures, Recommendation 1 (CI-1988). See T. J. Quinn, Metrologia 26, 69 (1989).
    [CrossRef]
  5. J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London) 277, 437 (1979).
    [CrossRef]
  6. J. H. Taylor and J. M. Weisberg, Astrophys. J. 345, 434 (1989).
    [CrossRef]
  7. M. M. Davis, J. H. Taylor, J. M. Weisberg, and D. C. Backer, Nature (London) 315, 547 (1985).
    [CrossRef]
  8. L. A. Rawley, J. H. Taylor, M. M. Davis, and D. W. Allan, Science 238, 761 (1987).
    [CrossRef] [PubMed]
  9. H. Lehmitz, J. Hattendorf-Ledwoch, R. Blatt, and H. Harde, Phys. Rev. Lett. 62, 2108 (1989).
    [CrossRef] [PubMed]
  10. W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Natl. Bur. Stand. (U.S.) Spec. Publ. 653(1983).
  11. P. Meystre and S. Stenholm, eds., feature on the mechanical effects of light, J. Opt. Soc. Am. B 2, 1705 ff. (1985).
    [CrossRef]
  12. S. Chu and C. Wieman, eds., feature on laser cooling and trapping of atoms, J. Opt. Soc. Am. B 6, 2019 ff. (1989).
  13. B. E. Blair, ed., Time and Frequency: Theory and Fundamentals, Natl. Bur. Stand. (U.S.) Monogr. 140(1974).
  14. D. Sullivan, D. Allan, D. Howe, and F. Walls, eds., Characterization of Clocks and Oscillators, Natl. Inst. Stand. Technol. Tech. Note 1337(1990).
  15. This and other possible atomic reference transitions are discussed by J. L. Hall, M. Zhu, and P. Buch, J. Opt. Soc. Am. B 6, 2194 (1989).
    [CrossRef]
  16. D. Middleton, Q. Appl. Math. 5, 445 (1948).
  17. D. Middleton, Q. Appl. Math. 7, 129 (1949).
  18. D. Middleton, Philos. Mag. 42, 689 (1951).
  19. D. Middleton, Q. Appl. Math. 9, 337 (1952).
  20. D. Middleton, Q. Appl. Math. 10, 35 (1952).
  21. There are a number of excellent reference books for the probability theory and stochastic processes. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  22. L. Mandel, J. Opt. Soc. Am. 71, 362 (1981).
    [CrossRef]
  23. N. Wiener, Acta Math. 55, 117 (1930).
    [CrossRef]
  24. A. Khintchine, Math. Ann. 109, 604 (1934).
    [CrossRef]
  25. G. C. Bjorklund, Opt. Lett. 5, 15 (1980).
    [CrossRef]
  26. J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
    [CrossRef]
  27. S. Rice, Bell Syst. Tech. J. 23, 282 (1944).
    [CrossRef]
  28. S. Rice, Bell Syst. Tech. J. 24, 46 (1945).
    [CrossRef]
  29. D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
    [CrossRef]
  30. E. T. Whittaker and G. N. Waston, Modern Analysis (Cambridge U. Press, Cambridge, 1927).
  31. M. Abramowitz and I. E. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Natl. Bur. Stand. Appl. Math. Ser. 55(1964).
  32. J. L. Hall, in International Conference on Quantum Optics, J. Harvey and D. Walls, eds. (Springer-Verlag, Berlin, 1986), and Ref. 6 therein.
  33. P. Juncar and J. Pinard, Opt. Commun. 14, 438 (1975).
    [CrossRef]
  34. P. Juncar and J. Pinard, Rev. Sci. Instrum. 53, 939 (1982).
    [CrossRef]
  35. M. Zhu and J. L. Hall, presented at the 14th International Quantum Electronics Conference, San Francisco, Calif., June 1986.
  36. J. L. Hall and M. Zhu, U.S. patent4,856,009 (August8, 1989).
  37. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  38. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, Opt. Lett. 17, 363 (1992).
    [CrossRef] [PubMed]
  39. D. Hils and J. L. Hall, Rev. Sci. Instrum. 58, 1406 (1987).
    [CrossRef]
  40. R. V. Pound, Rev. Sci. Instrum. 17, 490 (1946).
    [CrossRef] [PubMed]
  41. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
    [CrossRef]
  42. T. W. Hänsch and B. Couillaud, Opt. Commun. 35, 441 (1980).
    [CrossRef]
  43. J. L. Hall and T. W. Hänsch, Opt. Lett. 9, 502 (1984).
    [CrossRef] [PubMed]
  44. See, for example, R. D. Dorf, Modern Control Systems (Addison-Wesley, Reading, Mass., 1989).
  45. See, for example, R. Bellman and K. Cooke, Differential-Difference Equations (Academic, New York, 1963).
  46. One such commercially available software system is Simulab for the Macintosh computer; MathWorks Inc., Natick, Mass.
  47. Mention of a commercial product is for technical communication only. It does not imply endorsement, nor does it suggest that other products are necessarily less suitable for the application.
  48. To construct a stable, high-finesse Fabry–Perot cavity and to isolate it from the environmental disturbances is a sophisticated issue that is outside the scope of this paper.
  49. M. Zhu and J. L. Hall, “Short and long term stability of optical oscillators,” presented at the 1992 IEEE Frequency Control Symposium, Hershey, Pa., 1992.
  50. P. K. Runge and R. Rosenberg, IEEE J. Quantum Electron. QE-8, 910 (1972).
    [CrossRef]
  51. A. Schawlow and C. Townes, Phys. Rev. 112, 1940 (1958).
    [CrossRef]
  52. J. L. Hall, in International Conference on Lasers (Beijing, 1980), D. Wang, ed. (Wiley, New York, 1983).
  53. M. Sorem and A. Schawlow, Opt. Commun. 5, 148 (1972).
    [CrossRef]
  54. J. J. Synder and R. A. Keller, eds., feature on ultrasensitive laser spectroscopy, J. Opt. Soc. Am. 2, 1427 ff. (1985).
  55. M. Zhu, M. P. Winters, C. W. Oates, and J. L. Hall, “Sub-hertz stabilization of lasers using external phase and frequency modulators,” IEEE J. Quantum Electron. (to be published).
  56. A preliminary report on this experiment was presented previously: J. L. Hall, M. Zhu, F. Shimizu, and K. Shimizu, in Proceedings, 16th International Conference on Quantum Electronics, Tokyo, July 1988, H. Inaba, T. Yajima, and T. Ikegami, eds. (Japan Society of Applied Physics, Tokyo, 1988).
  57. D. Hils and J. L. Hall, in Frequency Standards and Metrology, A. De Marchi, ed. (Springer-Verlag, Berlin, 1989).
  58. Ch. Salomon, D. Hils, and J. L. Hall, J. Opt. Soc. Am. B 5, 1576 (1988).
    [CrossRef]
  59. The purpose of this amplitude stabilizer is to reduce the dye laser’s amplitude noise such that RV(0) is much smaller than Rφ(0) in Eq. (15). The reference He–Ne laser’s amplitude noise was not suppressed by any servo loop. This amplitude noise from the He–Ne laser turns out to be the main contribution to the residual amplitude (noise) modulation in the heterodyne signal. See the text for further discussion. Also see Ref. 32.
  60. S. Y. Wang and D. Bloom, Electron. Lett. 19, 554 (1983).
    [CrossRef]

1992 (1)

1991 (1)

J. L. Jespersen and D. W. Hanson eds., special issue on time and frequency, Proc. IEEE 79, 894 ff. (1991).
[CrossRef]

1990 (1)

D. Sullivan, D. Allan, D. Howe, and F. Walls, eds., Characterization of Clocks and Oscillators, Natl. Inst. Stand. Technol. Tech. Note 1337(1990).

1989 (5)

H. Lehmitz, J. Hattendorf-Ledwoch, R. Blatt, and H. Harde, Phys. Rev. Lett. 62, 2108 (1989).
[CrossRef] [PubMed]

S. Chu and C. Wieman, eds., feature on laser cooling and trapping of atoms, J. Opt. Soc. Am. B 6, 2019 ff. (1989).

Comité International des Poids et Mesures, Recommendation 1 (CI-1988). See T. J. Quinn, Metrologia 26, 69 (1989).
[CrossRef]

J. H. Taylor and J. M. Weisberg, Astrophys. J. 345, 434 (1989).
[CrossRef]

This and other possible atomic reference transitions are discussed by J. L. Hall, M. Zhu, and P. Buch, J. Opt. Soc. Am. B 6, 2194 (1989).
[CrossRef]

1988 (1)

1987 (2)

L. A. Rawley, J. H. Taylor, M. M. Davis, and D. W. Allan, Science 238, 761 (1987).
[CrossRef] [PubMed]

D. Hils and J. L. Hall, Rev. Sci. Instrum. 58, 1406 (1987).
[CrossRef]

1985 (3)

M. M. Davis, J. H. Taylor, J. M. Weisberg, and D. C. Backer, Nature (London) 315, 547 (1985).
[CrossRef]

P. Meystre and S. Stenholm, eds., feature on the mechanical effects of light, J. Opt. Soc. Am. B 2, 1705 ff. (1985).
[CrossRef]

J. J. Synder and R. A. Keller, eds., feature on ultrasensitive laser spectroscopy, J. Opt. Soc. Am. 2, 1427 ff. (1985).

1984 (2)

J. L. Hall and T. W. Hänsch, Opt. Lett. 9, 502 (1984).
[CrossRef] [PubMed]

Comité International des Poids et Mesures, Recommendation 1 (CI-1983). See “Editor’s note,” Metrologia 19, 163 (1984).

1983 (3)

W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Natl. Bur. Stand. (U.S.) Spec. Publ. 653(1983).

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

S. Y. Wang and D. Bloom, Electron. Lett. 19, 554 (1983).
[CrossRef]

1982 (2)

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[CrossRef]

P. Juncar and J. Pinard, Rev. Sci. Instrum. 53, 939 (1982).
[CrossRef]

1981 (2)

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
[CrossRef]

L. Mandel, J. Opt. Soc. Am. 71, 362 (1981).
[CrossRef]

1980 (2)

G. C. Bjorklund, Opt. Lett. 5, 15 (1980).
[CrossRef]

T. W. Hänsch and B. Couillaud, Opt. Commun. 35, 441 (1980).
[CrossRef]

1979 (1)

J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London) 277, 437 (1979).
[CrossRef]

1975 (1)

P. Juncar and J. Pinard, Opt. Commun. 14, 438 (1975).
[CrossRef]

1974 (1)

B. E. Blair, ed., Time and Frequency: Theory and Fundamentals, Natl. Bur. Stand. (U.S.) Monogr. 140(1974).

1972 (3)

J. L. Jespersen, B. E. Blair, and L. E. Gatterer, eds., special issue on time and frequency, Proc. IEEE 60, 476 ff. (1972).
[CrossRef]

P. K. Runge and R. Rosenberg, IEEE J. Quantum Electron. QE-8, 910 (1972).
[CrossRef]

M. Sorem and A. Schawlow, Opt. Commun. 5, 148 (1972).
[CrossRef]

1964 (1)

M. Abramowitz and I. E. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Natl. Bur. Stand. Appl. Math. Ser. 55(1964).

1958 (1)

A. Schawlow and C. Townes, Phys. Rev. 112, 1940 (1958).
[CrossRef]

1952 (2)

D. Middleton, Q. Appl. Math. 9, 337 (1952).

D. Middleton, Q. Appl. Math. 10, 35 (1952).

1951 (1)

D. Middleton, Philos. Mag. 42, 689 (1951).

1949 (1)

D. Middleton, Q. Appl. Math. 7, 129 (1949).

1948 (1)

D. Middleton, Q. Appl. Math. 5, 445 (1948).

1946 (1)

R. V. Pound, Rev. Sci. Instrum. 17, 490 (1946).
[CrossRef] [PubMed]

1945 (1)

S. Rice, Bell Syst. Tech. J. 24, 46 (1945).
[CrossRef]

1944 (1)

S. Rice, Bell Syst. Tech. J. 23, 282 (1944).
[CrossRef]

1934 (1)

A. Khintchine, Math. Ann. 109, 604 (1934).
[CrossRef]

1930 (1)

N. Wiener, Acta Math. 55, 117 (1930).
[CrossRef]

Allan, D. W.

L. A. Rawley, J. H. Taylor, M. M. Davis, and D. W. Allan, Science 238, 761 (1987).
[CrossRef] [PubMed]

Backer, D. C.

M. M. Davis, J. H. Taylor, J. M. Weisberg, and D. C. Backer, Nature (London) 315, 547 (1985).
[CrossRef]

Baer, T.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
[CrossRef]

Bellman, R.

See, for example, R. Bellman and K. Cooke, Differential-Difference Equations (Academic, New York, 1963).

Bjorklund, G. C.

Blatt, R.

H. Lehmitz, J. Hattendorf-Ledwoch, R. Blatt, and H. Harde, Phys. Rev. Lett. 62, 2108 (1989).
[CrossRef] [PubMed]

Bloom, D.

S. Y. Wang and D. Bloom, Electron. Lett. 19, 554 (1983).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Buch, P.

Cooke, K.

See, for example, R. Bellman and K. Cooke, Differential-Difference Equations (Academic, New York, 1963).

Couillaud, B.

T. W. Hänsch and B. Couillaud, Opt. Commun. 35, 441 (1980).
[CrossRef]

Davis, M. M.

L. A. Rawley, J. H. Taylor, M. M. Davis, and D. W. Allan, Science 238, 761 (1987).
[CrossRef] [PubMed]

M. M. Davis, J. H. Taylor, J. M. Weisberg, and D. C. Backer, Nature (London) 315, 547 (1985).
[CrossRef]

Dorf, R. D.

See, for example, R. D. Dorf, Modern Control Systems (Addison-Wesley, Reading, Mass., 1989).

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Elliott, D. S.

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Fowler, L. A.

J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London) 277, 437 (1979).
[CrossRef]

Hall, J. L.

This and other possible atomic reference transitions are discussed by J. L. Hall, M. Zhu, and P. Buch, J. Opt. Soc. Am. B 6, 2194 (1989).
[CrossRef]

Ch. Salomon, D. Hils, and J. L. Hall, J. Opt. Soc. Am. B 5, 1576 (1988).
[CrossRef]

D. Hils and J. L. Hall, Rev. Sci. Instrum. 58, 1406 (1987).
[CrossRef]

J. L. Hall and T. W. Hänsch, Opt. Lett. 9, 502 (1984).
[CrossRef] [PubMed]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
[CrossRef]

J. L. Hall, in International Conference on Quantum Optics, J. Harvey and D. Walls, eds. (Springer-Verlag, Berlin, 1986), and Ref. 6 therein.

M. Zhu and J. L. Hall, presented at the 14th International Quantum Electronics Conference, San Francisco, Calif., June 1986.

J. L. Hall and M. Zhu, U.S. patent4,856,009 (August8, 1989).

D. Hils and J. L. Hall, in Frequency Standards and Metrology, A. De Marchi, ed. (Springer-Verlag, Berlin, 1989).

M. Zhu, M. P. Winters, C. W. Oates, and J. L. Hall, “Sub-hertz stabilization of lasers using external phase and frequency modulators,” IEEE J. Quantum Electron. (to be published).

A preliminary report on this experiment was presented previously: J. L. Hall, M. Zhu, F. Shimizu, and K. Shimizu, in Proceedings, 16th International Conference on Quantum Electronics, Tokyo, July 1988, H. Inaba, T. Yajima, and T. Ikegami, eds. (Japan Society of Applied Physics, Tokyo, 1988).

M. Zhu and J. L. Hall, “Short and long term stability of optical oscillators,” presented at the 1992 IEEE Frequency Control Symposium, Hershey, Pa., 1992.

J. L. Hall, in International Conference on Lasers (Beijing, 1980), D. Wang, ed. (Wiley, New York, 1983).

Hänsch, T. W.

J. L. Hall and T. W. Hänsch, Opt. Lett. 9, 502 (1984).
[CrossRef] [PubMed]

T. W. Hänsch and B. Couillaud, Opt. Commun. 35, 441 (1980).
[CrossRef]

Harde, H.

H. Lehmitz, J. Hattendorf-Ledwoch, R. Blatt, and H. Harde, Phys. Rev. Lett. 62, 2108 (1989).
[CrossRef] [PubMed]

Hattendorf-Ledwoch, J.

H. Lehmitz, J. Hattendorf-Ledwoch, R. Blatt, and H. Harde, Phys. Rev. Lett. 62, 2108 (1989).
[CrossRef] [PubMed]

Hils, D.

Ch. Salomon, D. Hils, and J. L. Hall, J. Opt. Soc. Am. B 5, 1576 (1988).
[CrossRef]

D. Hils and J. L. Hall, Rev. Sci. Instrum. 58, 1406 (1987).
[CrossRef]

D. Hils and J. L. Hall, in Frequency Standards and Metrology, A. De Marchi, ed. (Springer-Verlag, Berlin, 1989).

Hollberg, L.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Juncar, P.

P. Juncar and J. Pinard, Rev. Sci. Instrum. 53, 939 (1982).
[CrossRef]

P. Juncar and J. Pinard, Opt. Commun. 14, 438 (1975).
[CrossRef]

Khintchine, A.

A. Khintchine, Math. Ann. 109, 604 (1934).
[CrossRef]

Kimble, H. J.

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Lalezari, R.

Lehmitz, H.

H. Lehmitz, J. Hattendorf-Ledwoch, R. Blatt, and H. Harde, Phys. Rev. Lett. 62, 2108 (1989).
[CrossRef] [PubMed]

Mandel, L.

McCulloch, P. M.

J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London) 277, 437 (1979).
[CrossRef]

Middleton, D.

D. Middleton, Q. Appl. Math. 9, 337 (1952).

D. Middleton, Q. Appl. Math. 10, 35 (1952).

D. Middleton, Philos. Mag. 42, 689 (1951).

D. Middleton, Q. Appl. Math. 7, 129 (1949).

D. Middleton, Q. Appl. Math. 5, 445 (1948).

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Oates, C. W.

M. Zhu, M. P. Winters, C. W. Oates, and J. L. Hall, “Sub-hertz stabilization of lasers using external phase and frequency modulators,” IEEE J. Quantum Electron. (to be published).

Papoulis, A.

There are a number of excellent reference books for the probability theory and stochastic processes. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Pinard, J.

P. Juncar and J. Pinard, Rev. Sci. Instrum. 53, 939 (1982).
[CrossRef]

P. Juncar and J. Pinard, Opt. Commun. 14, 438 (1975).
[CrossRef]

Pound, R. V.

R. V. Pound, Rev. Sci. Instrum. 17, 490 (1946).
[CrossRef] [PubMed]

Quinn, T. J.

Comité International des Poids et Mesures, Recommendation 1 (CI-1988). See T. J. Quinn, Metrologia 26, 69 (1989).
[CrossRef]

Rawley, L. A.

L. A. Rawley, J. H. Taylor, M. M. Davis, and D. W. Allan, Science 238, 761 (1987).
[CrossRef] [PubMed]

Rempe, G.

Rice, S.

S. Rice, Bell Syst. Tech. J. 24, 46 (1945).
[CrossRef]

S. Rice, Bell Syst. Tech. J. 23, 282 (1944).
[CrossRef]

Robinson, H. G.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
[CrossRef]

Rosenberg, R.

P. K. Runge and R. Rosenberg, IEEE J. Quantum Electron. QE-8, 910 (1972).
[CrossRef]

Roy, R.

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[CrossRef]

Runge, P. K.

P. K. Runge and R. Rosenberg, IEEE J. Quantum Electron. QE-8, 910 (1972).
[CrossRef]

Salomon, Ch.

Schawlow, A.

M. Sorem and A. Schawlow, Opt. Commun. 5, 148 (1972).
[CrossRef]

A. Schawlow and C. Townes, Phys. Rev. 112, 1940 (1958).
[CrossRef]

Shimizu, F.

A preliminary report on this experiment was presented previously: J. L. Hall, M. Zhu, F. Shimizu, and K. Shimizu, in Proceedings, 16th International Conference on Quantum Electronics, Tokyo, July 1988, H. Inaba, T. Yajima, and T. Ikegami, eds. (Japan Society of Applied Physics, Tokyo, 1988).

Shimizu, K.

A preliminary report on this experiment was presented previously: J. L. Hall, M. Zhu, F. Shimizu, and K. Shimizu, in Proceedings, 16th International Conference on Quantum Electronics, Tokyo, July 1988, H. Inaba, T. Yajima, and T. Ikegami, eds. (Japan Society of Applied Physics, Tokyo, 1988).

Smith, S. J.

D. S. Elliott, R. Roy, and S. J. Smith, Phys. Rev. A 26, 12 (1982).
[CrossRef]

Sorem, M.

M. Sorem and A. Schawlow, Opt. Commun. 5, 148 (1972).
[CrossRef]

Taylor, J. H.

J. H. Taylor and J. M. Weisberg, Astrophys. J. 345, 434 (1989).
[CrossRef]

L. A. Rawley, J. H. Taylor, M. M. Davis, and D. W. Allan, Science 238, 761 (1987).
[CrossRef] [PubMed]

M. M. Davis, J. H. Taylor, J. M. Weisberg, and D. C. Backer, Nature (London) 315, 547 (1985).
[CrossRef]

J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London) 277, 437 (1979).
[CrossRef]

Thompson, R. J.

Townes, C.

A. Schawlow and C. Townes, Phys. Rev. 112, 1940 (1958).
[CrossRef]

Wang, S. Y.

S. Y. Wang and D. Bloom, Electron. Lett. 19, 554 (1983).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983).
[CrossRef]

Waston, G. N.

E. T. Whittaker and G. N. Waston, Modern Analysis (Cambridge U. Press, Cambridge, 1927).

Weisberg, J. M.

J. H. Taylor and J. M. Weisberg, Astrophys. J. 345, 434 (1989).
[CrossRef]

M. M. Davis, J. H. Taylor, J. M. Weisberg, and D. C. Backer, Nature (London) 315, 547 (1985).
[CrossRef]

Whittaker, E. T.

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Other (17)

E. T. Whittaker and G. N. Waston, Modern Analysis (Cambridge U. Press, Cambridge, 1927).

M. Zhu and J. L. Hall, presented at the 14th International Quantum Electronics Conference, San Francisco, Calif., June 1986.

J. L. Hall and M. Zhu, U.S. patent4,856,009 (August8, 1989).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).

J. L. Hall, in International Conference on Quantum Optics, J. Harvey and D. Walls, eds. (Springer-Verlag, Berlin, 1986), and Ref. 6 therein.

There are a number of excellent reference books for the probability theory and stochastic processes. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

J. L. Hall, in International Conference on Lasers (Beijing, 1980), D. Wang, ed. (Wiley, New York, 1983).

M. Zhu, M. P. Winters, C. W. Oates, and J. L. Hall, “Sub-hertz stabilization of lasers using external phase and frequency modulators,” IEEE J. Quantum Electron. (to be published).

A preliminary report on this experiment was presented previously: J. L. Hall, M. Zhu, F. Shimizu, and K. Shimizu, in Proceedings, 16th International Conference on Quantum Electronics, Tokyo, July 1988, H. Inaba, T. Yajima, and T. Ikegami, eds. (Japan Society of Applied Physics, Tokyo, 1988).

D. Hils and J. L. Hall, in Frequency Standards and Metrology, A. De Marchi, ed. (Springer-Verlag, Berlin, 1989).

See, for example, R. D. Dorf, Modern Control Systems (Addison-Wesley, Reading, Mass., 1989).

See, for example, R. Bellman and K. Cooke, Differential-Difference Equations (Academic, New York, 1963).

One such commercially available software system is Simulab for the Macintosh computer; MathWorks Inc., Natick, Mass.

Mention of a commercial product is for technical communication only. It does not imply endorsement, nor does it suggest that other products are necessarily less suitable for the application.

To construct a stable, high-finesse Fabry–Perot cavity and to isolate it from the environmental disturbances is a sophisticated issue that is outside the scope of this paper.

M. Zhu and J. L. Hall, “Short and long term stability of optical oscillators,” presented at the 1992 IEEE Frequency Control Symposium, Hershey, Pa., 1992.

The purpose of this amplitude stabilizer is to reduce the dye laser’s amplitude noise such that RV(0) is much smaller than Rφ(0) in Eq. (15). The reference He–Ne laser’s amplitude noise was not suppressed by any servo loop. This amplitude noise from the He–Ne laser turns out to be the main contribution to the residual amplitude (noise) modulation in the heterodyne signal. See the text for further discussion. Also see Ref. 32.

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Figures (15)

Fig. 1
Fig. 1

Contours for Eq. (4). Closure is (a) in the upper half-plane for VN(t) ≥ −1 and. (b) in the lower half-plane for VN(t) < −1. See text for details.

Fig. 2
Fig. 2

Detected signal versus detuning in the reflection scheme for a, dispersion and b, absorption. The modulation frequency is ~48 times the cavity linewidth (HWHM). The detuning shown is expressed in terms of cavity linewidth (HWHM).

Fig. 3
Fig. 3

Responses of a Fabry–Perot cavity when the Pound–Drever–Hall scheme is used in the dispersion mode with zero de-tuning: a, transient response to a step phase disturbance; b, the transient response to a step frequency disturbance; c, relative gain of the steady response to FM noise; d, phase delay of the steady response to FM noise. In c and d the crosses are the measured data, while the solid curves are theoretical curves. Cavity linewidth (HWHM) is 6 kHz. The modulation frequency is 12.5 MHz with modulation index ~0.9.

Fig. 4
Fig. 4

Heterodyne detection scheme. A beam splitter (B.S.) combines the reference laser beam with the test laser’s beam. A photodetector (P.D.) detects the heterodyne signal. See the text for further discussion.

Fig. 5
Fig. 5

Plot of heterodyne signal between the commercial dye laser, which is locked to its own reference cavity, and the He–Ne laser at 612 nm. The crosses are the experimental data. The solid curve is a Gaussian profile fit to the data. The effective linewidth (FWHM) is 450 kHz.

Fig. 6
Fig. 6

Block diagram of the external optical phase/frequency stabilizer. See the text for details.

Fig. 7
Fig. 7

Block diagram for locking the dye laser’s frequency to a high-finesse Fabry–Perot cavity. See description in text.

Fig. 8
Fig. 8

Result of cavity locking. The resolution bandwidth is 1 kHz; gain increases 10 dB/step. At the highest gain, no intrinsic laser noise remains, and even the shot noise below 100 kHz has been written as an equivalent FM onto the laser. The dashed line shows the shot noise level at 1-kHz resolution bandwidth.

Fig. 9
Fig. 9

Block diagram for optical phase locking the dye laser to a reference He–Ne laser at 612 nm. See text for details.

Fig. 10
Fig. 10

a, Histogram of the residual phase fluctuations φN′(t) in Eq. (38). b, Histogram of the residual amplitude fluctuations VN′(t) in Eq. (38). Solid curves are the Gaussian distributions fitted to the data.

Fig. 11
Fig. 11

Correlation coefficient of the phase fluctuations and amplitude fluctuations as defined by Eq. (45). Basically the amplitude and frequency fluctuations are uncorrelated. See text.

Fig. 12
Fig. 12

Power spectral densities of a, the residual phase fluctuations and, b, the residual amplitude fluctuations. In a the increased noise near 200 kHz results mainly from strong excitation by the argon-ion laser, combined with imperfect crossover between the AOM and EOM transducers. In a, 0 dB corresponds to the digitizing error, which is ~1.16 × 10−13 rad2/Hz; in b, 0 dB represents the digitizing error, which is ~1.16 × 10−13/Hz.

Fig. 13
Fig. 13

Normalized autocorrelation functions of a, residual phase fluctuations, and b, the residual amplitude fluctuations under phase-locked conditions. Finite high-frequency gain and some remaining settling problems in the AOM–EOM crossover region lead to some residual phase errors. Still, 97% of the emission is captured by the optical phase-locked loop. See text.

Fig. 14
Fig. 14

Solid curve, the noise power spectral density reconstructed by using the phase and amplitude fluctuations. The dotted curve is the power spectral density of the heterodyne signal taken by the rf spectrum analyzer when the dye laser is phase locked to the reference He–Ne laser. The linewidth of the central peak is limited by the linewidth of the built-in filter of the rf spectrum analyzer. The resolution bandwidth of this display is 30 kHz.

Fig. 15
Fig. 15

Integral distribution of noise power: Noise power fraction within the frequency interval |ωω0|/2π. Note that the total noise power is only 3% of the carrier in our experiment.

Equations (45)

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E ( t ) = e ^ E ( t ) = e ^ E 0 exp ( - i ω 0 t - i φ 0 ) ,
E ( t ) = ( t ) exp [ - i ω 0 t - i φ N ( t ) ] ,
ω ( t ) = ω 0 + d φ N ( t ) d t .
( t ) = - E 0 2 π C d ζ ζ 2 exp { i ζ [ 1 + V N ( t ) ] } = { E 0 [ 1 + V N ( t ) ] V N ( t ) - 1 0 V N ( t ) < - 1 ,
R X ( τ ) X ( t ) X * ( t + τ ) ,
P E ( ω ) = 1 2 π - R E ( τ ) exp ( - i ω τ ) d τ .
R E ( τ ) = E ( t ) E * ( t + τ ) = E 0 2 4 π 2 exp ( i ω 0 τ ) C d ζ C * d η 1 ζ 2 η 2 exp { i ζ [ 1 + V N ( t ) ] - i η [ 1 + V N ( t + τ ) ] - i φ N ( t ) + i φ N ( t + τ ) } = E 0 2 4 π 2 exp ( i ω 0 τ ) C d ζ C * d η exp ( i ζ - i η ) ζ 2 η 2 exp [ i ζ V N ( t ) - i η V N ( t + τ ) - i φ N ( t ) + i φ N ( t + τ ) ] = E 0 2 4 π 2 exp ( i ω 0 τ ) C d ζ C * d η exp ( i ζ - i η ) ζ 2 η 2 × Φ ( ζ , - η , - 1 , 1 ; τ ) ,
Φ ( ζ , η , ξ , σ ; τ ) = exp [ i ζ V N ( t ) + i η V N ( t + τ ) + i ξ φ N ( t ) + i σ φ N ( t + τ ) ]
V N ( t ) = κ φ N ( t + t 0 ) + U N ( t ) ,
Φ ( ζ , η , ξ , σ , τ ) = exp [ i ζ V N ( t ) + i η V N ( t + τ ) + i ξ φ N ( t ) + i σ φ N ( t + τ ) ] = exp [ i ζ U N ( t ) + i η U N ( t + τ ) ] exp [ i ζ κ φ N ( t + t 0 ) + i η κ φ N ( t + t 0 + τ ) + i ξ φ N ( t ) + i σ φ N ( t + τ ) ] .
R E ( τ ) = E 0 2 exp ( i ω 0 τ ) { 1 + V N ( t ) V N ( t + τ ) } × exp [ i φ N ( t + τ ) - i φ N ( t ) ] .
R V ( τ ) = V N ( t ) V N ( t + τ ) ,
R φ ( τ ) = φ N ( t ) φ N ( t + τ )
exp [ i φ N ( t + τ ) - i φ N ( t ) ] = exp [ R φ ( τ ) - R φ ( 0 ) ] ,
R E ( τ ) = E 0 2 exp ( i ω 0 τ ) exp { - Ω φ ( τ ) } + E 0 2 exp ( i ω 0 τ ) R V ( τ ) exp [ - Ω φ ( τ ) ] ,
Ω φ ( τ ) R φ ( 0 ) - R φ ( τ ) .
P total = - P E ( ω ) d ω = R E ( 0 ) = E 0 2 { 1 + R V ( 0 ) } .
R V ( τ ) τ = 0 ,
Ω φ ( τ ) R φ ( 0 ) - R φ ( τ ) = - d ω P φ ( ω ) [ 1 - exp ( i ω τ ) ] ,
P φ ( ω ) = { P Noise ( ω ) for phase modulation P Noise ( ω ) / ω 2 for frequency modulation
R E ( τ ) τ = E 0 2 exp [ - R φ ( 0 ) ] exp ( i ω 0 τ ) = E 0 2 exp ( - φ rms 2 ) exp ( i ω 0 τ ) ,
R E ( τ ) = E 0 2 exp ( i ω 0 τ ) exp ( - φ rms 2 ) + E 0 2 exp ( i ω 0 τ ) × exp ( - φ rms 2 ) { [ 1 + R V ( τ ) ] exp [ R φ ( τ ) ] - 1 }
P Noise ( ω ) = C 0 + C 1 ω + C 2 ω 2 + C 3 ω 3 + ,
Ω φ ( τ ) τ = O ( C 0 ω - C 1 ln ω - C 2 ω + ) ω 0 + ,
P Noise ( ω ) = { π f N 2 B ω 2 π B 0 ω > 2 π B ,
Ω φ ( τ ) = - d ω P Noise ( ω ) ω 2 [ 1 - exp ( i ω τ ) ] = 2 π τ f N 2 B 0 π B τ d ξ sin 2 ξ ξ 2 .
P E ( ω ) = E 0 2 ( 8 π 3 f N 2 ) 1 / 2 exp [ - ( ω - ω 0 ) 2 8 π 2 f N 2 ] .
P E ( ω ) = E 0 2 π π 2 f N 2 B ( ω - ω 0 ) 2 + ( π 2 f N 2 B ) 2 .
P ω ( ω ) closed loop = P ω ( ω ) open loop 1 + G ( ω ) 2 ,
P ω ( ω ) closed loop = b π ω 2 ω i 2 + ( K p + 1 ) 2 ω 2 ,
Ω φ ( τ ) = - P ω ( ω ) closed loop ω 2 [ 1 - exp ( i ω τ ) ] d ω = b ( K p + 1 ) ω i [ 1 - exp ( - ω i τ K p + 1 ) ] = b [ 1 - exp ( - ω i τ K p + 1 ) ] ,
b b ( K p + 1 ) ω i .
P E ( ω ) = 1 2 π - R E ( τ ) exp ( - i ω τ ) d τ = 1 2 π - E 0 2 exp [ - Ω φ ( τ ) ] exp [ - i ( ω - ω 0 ) τ ] d τ = E 0 2 2 π exp ( - b ) - d τ × exp { b exp [ - ω i τ K p + 1 ] - i ( ω - ω 0 ) τ } .
θ ( K p + 1 ) ω - ω 0 ω i ,
P E ( ω ) = E 0 2 exp ( - b ) δ ( ω - ω 0 ) + E 0 2 2 π b ω i 2 exp ( - b ) [ F 2 2 ( 1 , 1 - i θ ; 2 , 2 - i θ ; b ) 1 - i θ + F 2 2 ( 1 , 1 + i θ ; 2 , 2 + i θ ; b ) 1 + i θ ] = E 0 2 exp ( - b ) δ ( ω - ω 0 ) + E 0 2 2 π exp ( - b ) n = 1 b n n ! n ω i K p + 1 ( ω - ω 0 ) 2 + ( n ω i K p + 1 ) 2 .
F 2 2 ( α ; β ; γ ; δ ; z ) = n = 0 1 n ! ( α ) n ( β ) n ( γ ) n ( δ ) n z n
E r ( t ) = E r [ 1 + V r ( t ) ] exp [ - i ω r t - i φ r ( t ) ] ,
I photo = η E 0 E r [ 1 + V N ( t ) ] [ 1 + V r ( t ) ] × cos [ ( ω 0 - ω r ) t + φ N ( t ) - φ r ( t ) ] η E 0 E r [ 1 + V N ( t ) ] cos [ Δ t + φ N ( t ) ] ,
V N ( t ) V N ( t ) + V r ( t ) + V N ( t ) V r ( t ) ,
φ N ( t ) φ N ( t ) - φ r ( t ) ,
Δ ω 0 - ω r ,
ω i τ ( 1 - K p 2 ) 1 / 2 < π 2 + sin - 1 K p ,
0 K p < 1
( K p + 1 ) ω i τ < ( K p + 1 ) ( π / 2 + sin - 1 K p ) ( 1 - K p 2 ) 1 / 2
C ( τ ) = V N ( t ) φ N ( t + τ ) { [ V N ( t ) ] 2 [ φ N ( t ) ] 2 } 1 / 2 ,

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