Abstract

Homogeneous laser radiation in the visible or infrared range can excite high-order subharmonics at the cyclotron frequency of free electrons in the millimeter or microwave range. This may provide coherent links between lasers and rf or microwave frequency standards. In order to divide the frequency of a CO2 laser (λ ≈ 10 μm) by a factor of 100 down to λ ≈ 1 mm in one step, a cw laser power as low as 10−6 W is sufficient.

© 1987 Optical Society of America

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References

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  1. A. E. Kaplan, Phys. Rev. Lett. 48, 138 (1982).
    [Crossref]
  2. G. Gabrielse, H. Dehmelt, W. Kells, Phys. Rev. Lett. 54, 537 (1985).
    [Crossref] [PubMed]
  3. A. E. Kaplan, Phys. Rev. Lett. 56, 456 (1986).
    [Crossref] [PubMed]
  4. A. E. Kaplan, IEEE J. Quantum Electron. QE-24, 1544 (1985);Nature 317, 476 (1985).
    [Crossref]
  5. E. M. Macmillan, Phys. Rev. 68, 143 (1945);V. Veksler, J. Phys. (USSR) 9, 153 (1945).
    [Crossref]
  6. D. Bohm, L. L. Foldy, Phys. Rev. 72, 649 (1947).
    [Crossref]
  7. D. J. Wineland, J. Appl. Phys. 50, 2528 (1979);J. C. Bergquist, D. J. Wineland in Proceedings of the 33rd Annual Symposium on Frequency Control (American Institute of Physics, New York, 1979).
    [Crossref]
  8. L. D. Landau, E. M. Lifshits, The Classical Theory of Fields (Addison-Wesley, Cambridge, Mass., 1951).
  9. J. Schwinger, Phys. Rev. 75, 1912 (1949).
    [Crossref]
  10. A. E. Kaplan, Radio Eng. Electron. Phys. 8, 1340 (1963);Radio Eng. Electron. Phys. 9, 1424 (1964);Radio Eng. Electron. Phys. 11, 1214, 1354 (1966);A. E. Kaplan, Yu. A. Kravtsov, V. A. Rylov, Parametric Oscillators and Frequency Dividers (Soviet Radio, Moscow, 1966; in Russian).
  11. A. E. Kaplan, Radiophys. Quantum Electron. 11, 900 (1968).
    [Crossref]
  12. K. M. Evenson, D. A. Jennings, F. R. Peterson, J. S. Wells, in Proceedings of the Third International Conference on Laser Spectroscopy, J. L. Hall, J. L. Carlsten, eds. (Springer-Verlag, Heidelberg, 1977), Vol. 7, p. 56;D. J. E. Knight, P. T. Woods, J. Phys. E. 9, 898 (1976).
    [Crossref]
  13. D. A. Jennings, C. R. Pollack, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, Opt. Lett. 8, 136 (1983).
    [Crossref] [PubMed]
  14. R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
    [Crossref]
  15. It is worth noting that a standing-wave configuration is used in this Letter for the sake of symmetry and calculation simplicity. In fact, subharmonic oscillations can be obtained by using a plane traveling wave instead of a standing wave.
  16. See, e.g., A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
    [Crossref]

1986 (1)

A. E. Kaplan, Phys. Rev. Lett. 56, 456 (1986).
[Crossref] [PubMed]

1985 (3)

A. E. Kaplan, IEEE J. Quantum Electron. QE-24, 1544 (1985);Nature 317, 476 (1985).
[Crossref]

G. Gabrielse, H. Dehmelt, W. Kells, Phys. Rev. Lett. 54, 537 (1985).
[Crossref] [PubMed]

See, e.g., A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[Crossref]

1984 (1)

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[Crossref]

1983 (1)

1982 (1)

A. E. Kaplan, Phys. Rev. Lett. 48, 138 (1982).
[Crossref]

1979 (1)

D. J. Wineland, J. Appl. Phys. 50, 2528 (1979);J. C. Bergquist, D. J. Wineland in Proceedings of the 33rd Annual Symposium on Frequency Control (American Institute of Physics, New York, 1979).
[Crossref]

1968 (1)

A. E. Kaplan, Radiophys. Quantum Electron. 11, 900 (1968).
[Crossref]

1963 (1)

A. E. Kaplan, Radio Eng. Electron. Phys. 8, 1340 (1963);Radio Eng. Electron. Phys. 9, 1424 (1964);Radio Eng. Electron. Phys. 11, 1214, 1354 (1966);A. E. Kaplan, Yu. A. Kravtsov, V. A. Rylov, Parametric Oscillators and Frequency Dividers (Soviet Radio, Moscow, 1966; in Russian).

1949 (1)

J. Schwinger, Phys. Rev. 75, 1912 (1949).
[Crossref]

1947 (1)

D. Bohm, L. L. Foldy, Phys. Rev. 72, 649 (1947).
[Crossref]

1945 (1)

E. M. Macmillan, Phys. Rev. 68, 143 (1945);V. Veksler, J. Phys. (USSR) 9, 153 (1945).
[Crossref]

Bohm, D.

D. Bohm, L. L. Foldy, Phys. Rev. 72, 649 (1947).
[Crossref]

Brewer, R. G.

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[Crossref]

Dehmelt, H.

G. Gabrielse, H. Dehmelt, W. Kells, Phys. Rev. Lett. 54, 537 (1985).
[Crossref] [PubMed]

DeVoe, R. G.

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[Crossref]

Drullinger, R. E.

Evenson, K. M.

D. A. Jennings, C. R. Pollack, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, Opt. Lett. 8, 136 (1983).
[Crossref] [PubMed]

K. M. Evenson, D. A. Jennings, F. R. Peterson, J. S. Wells, in Proceedings of the Third International Conference on Laser Spectroscopy, J. L. Hall, J. L. Carlsten, eds. (Springer-Verlag, Heidelberg, 1977), Vol. 7, p. 56;D. J. E. Knight, P. T. Woods, J. Phys. E. 9, 898 (1976).
[Crossref]

Foldy, L. L.

D. Bohm, L. L. Foldy, Phys. Rev. 72, 649 (1947).
[Crossref]

Gabrielse, G.

G. Gabrielse, H. Dehmelt, W. Kells, Phys. Rev. Lett. 54, 537 (1985).
[Crossref] [PubMed]

Jennings, D. A.

D. A. Jennings, C. R. Pollack, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, Opt. Lett. 8, 136 (1983).
[Crossref] [PubMed]

K. M. Evenson, D. A. Jennings, F. R. Peterson, J. S. Wells, in Proceedings of the Third International Conference on Laser Spectroscopy, J. L. Hall, J. L. Carlsten, eds. (Springer-Verlag, Heidelberg, 1977), Vol. 7, p. 56;D. J. E. Knight, P. T. Woods, J. Phys. E. 9, 898 (1976).
[Crossref]

Kaplan, A. E.

A. E. Kaplan, Phys. Rev. Lett. 56, 456 (1986).
[Crossref] [PubMed]

A. E. Kaplan, IEEE J. Quantum Electron. QE-24, 1544 (1985);Nature 317, 476 (1985).
[Crossref]

See, e.g., A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[Crossref]

A. E. Kaplan, Phys. Rev. Lett. 48, 138 (1982).
[Crossref]

A. E. Kaplan, Radiophys. Quantum Electron. 11, 900 (1968).
[Crossref]

A. E. Kaplan, Radio Eng. Electron. Phys. 8, 1340 (1963);Radio Eng. Electron. Phys. 9, 1424 (1964);Radio Eng. Electron. Phys. 11, 1214, 1354 (1966);A. E. Kaplan, Yu. A. Kravtsov, V. A. Rylov, Parametric Oscillators and Frequency Dividers (Soviet Radio, Moscow, 1966; in Russian).

Kells, W.

G. Gabrielse, H. Dehmelt, W. Kells, Phys. Rev. Lett. 54, 537 (1985).
[Crossref] [PubMed]

Landau, L. D.

L. D. Landau, E. M. Lifshits, The Classical Theory of Fields (Addison-Wesley, Cambridge, Mass., 1951).

Law, C. T.

See, e.g., A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[Crossref]

Lifshits, E. M.

L. D. Landau, E. M. Lifshits, The Classical Theory of Fields (Addison-Wesley, Cambridge, Mass., 1951).

Macmillan, E. M.

E. M. Macmillan, Phys. Rev. 68, 143 (1945);V. Veksler, J. Phys. (USSR) 9, 153 (1945).
[Crossref]

Peterson, F. R.

D. A. Jennings, C. R. Pollack, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, Opt. Lett. 8, 136 (1983).
[Crossref] [PubMed]

K. M. Evenson, D. A. Jennings, F. R. Peterson, J. S. Wells, in Proceedings of the Third International Conference on Laser Spectroscopy, J. L. Hall, J. L. Carlsten, eds. (Springer-Verlag, Heidelberg, 1977), Vol. 7, p. 56;D. J. E. Knight, P. T. Woods, J. Phys. E. 9, 898 (1976).
[Crossref]

Pollack, C. R.

Schwinger, J.

J. Schwinger, Phys. Rev. 75, 1912 (1949).
[Crossref]

Wells, J. S.

D. A. Jennings, C. R. Pollack, F. R. Peterson, R. E. Drullinger, K. M. Evenson, J. S. Wells, Opt. Lett. 8, 136 (1983).
[Crossref] [PubMed]

K. M. Evenson, D. A. Jennings, F. R. Peterson, J. S. Wells, in Proceedings of the Third International Conference on Laser Spectroscopy, J. L. Hall, J. L. Carlsten, eds. (Springer-Verlag, Heidelberg, 1977), Vol. 7, p. 56;D. J. E. Knight, P. T. Woods, J. Phys. E. 9, 898 (1976).
[Crossref]

Wineland, D. J.

D. J. Wineland, J. Appl. Phys. 50, 2528 (1979);J. C. Bergquist, D. J. Wineland in Proceedings of the 33rd Annual Symposium on Frequency Control (American Institute of Physics, New York, 1979).
[Crossref]

IEEE J. Quantum Electron. (2)

A. E. Kaplan, IEEE J. Quantum Electron. QE-24, 1544 (1985);Nature 317, 476 (1985).
[Crossref]

See, e.g., A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[Crossref]

J. Appl. Phys. (1)

D. J. Wineland, J. Appl. Phys. 50, 2528 (1979);J. C. Bergquist, D. J. Wineland in Proceedings of the 33rd Annual Symposium on Frequency Control (American Institute of Physics, New York, 1979).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (3)

E. M. Macmillan, Phys. Rev. 68, 143 (1945);V. Veksler, J. Phys. (USSR) 9, 153 (1945).
[Crossref]

D. Bohm, L. L. Foldy, Phys. Rev. 72, 649 (1947).
[Crossref]

J. Schwinger, Phys. Rev. 75, 1912 (1949).
[Crossref]

Phys. Rev. A (1)

R. G. DeVoe, R. G. Brewer, Phys. Rev. A 30, 2827 (1984).
[Crossref]

Phys. Rev. Lett. (3)

A. E. Kaplan, Phys. Rev. Lett. 48, 138 (1982).
[Crossref]

G. Gabrielse, H. Dehmelt, W. Kells, Phys. Rev. Lett. 54, 537 (1985).
[Crossref] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 56, 456 (1986).
[Crossref] [PubMed]

Radio Eng. Electron. Phys. (1)

A. E. Kaplan, Radio Eng. Electron. Phys. 8, 1340 (1963);Radio Eng. Electron. Phys. 9, 1424 (1964);Radio Eng. Electron. Phys. 11, 1214, 1354 (1966);A. E. Kaplan, Yu. A. Kravtsov, V. A. Rylov, Parametric Oscillators and Frequency Dividers (Soviet Radio, Moscow, 1966; in Russian).

Radiophys. Quantum Electron. (1)

A. E. Kaplan, Radiophys. Quantum Electron. 11, 900 (1968).
[Crossref]

Other (3)

K. M. Evenson, D. A. Jennings, F. R. Peterson, J. S. Wells, in Proceedings of the Third International Conference on Laser Spectroscopy, J. L. Hall, J. L. Carlsten, eds. (Springer-Verlag, Heidelberg, 1977), Vol. 7, p. 56;D. J. E. Knight, P. T. Woods, J. Phys. E. 9, 898 (1976).
[Crossref]

It is worth noting that a standing-wave configuration is used in this Letter for the sake of symmetry and calculation simplicity. In fact, subharmonic oscillations can be obtained by using a plane traveling wave instead of a standing wave.

L. D. Landau, E. M. Lifshits, The Classical Theory of Fields (Addison-Wesley, Cambridge, Mass., 1951).

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Figures (3)

Fig. 1
Fig. 1

Extremum kinetic energy γ − 1 of the cyclotron motion versus driving parameter μ for various orders of subharmonics n. Insert: incident light configuration with respect to the dc magnetic field H0.

Fig. 2
Fig. 2

Kinetic energy of the cyclotron motion γ − 1 versus the dimensionless frequency of a driving field ω0 for various orders of subharmonics n (n = 1 is the main resonance, n = 2 is the second subharmonic, etc.) and for the fixed amplitude parameter μ = f/Γ = E/E0 of a driving field. a, μ = 0.9; b, μ = 3.5; c, μ = 16. The thick solid branches correspond to stable states; the thick broken branches to unstable states. All the branches are stretched out along lines determined by the formula γ = nΩ0/ω for each order n.

Fig. 3
Fig. 3

Threshold driving parameter μth = Eth/E0 (curve 1) and kinetic energy γ − 1 corresponding to μth (curve 2) versus order of subharmonic n. Insert: magnified plot in the vicinity of the origin.

Equations (9)

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E j = ê y E cos [ ω t + ( 1 ) j ( k x ψ ) ] , j = 1 , 2 , k = ω / c ,
Ω 0 1 d ρ / d t + Γ γ ρ = ê y ( f 1 + f 2 ) + γ 1 [ ρ × ê z ] × [ 1 + ( f 1 f 2 ) + g ( ζ ) ,
ζ ( t ) = ζ c [ ê x sin ( Ω t + ϕ ) + ê y cos ( Ω t + ϕ ) ] + ζ ( 0 ) ,
1 Ω 0 d ζ c d t + Γ γ ζ c = f sin ( n ϕ ψ ) × [ J n 1 ( n β ) J n + 1 ( n β ) ] ,
1 Ω 0 d ϕ d t + ( ω n Ω 0 1 γ ) = 2 β 2 γ 3 f cos ( n ϕ ψ ) J n ( n β ) ,
γ = n Ω 0 / ω + Δ γ ( and ζ c = ρ = γ 1 γ 2 ) ,
Δ γ Γ = ± 2 J n ( n β ) β 2 γ { μ 2 β 2 γ 4 [ J n 1 ( n β ) J n + 1 ( n β ) ] 2 } 1 / 2 ,
μ = β / ( 1 β 2 ) [ J n 1 ( n β ) J n + 1 ( n β ) ] ,
E 0 = 2 e Ω 0 2 / 3 c 2 = ( 2 / 3 ) ( m 0 c 2 / e ) k 0 2 r e ,

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