Abstract

Novel spectral line shapes of optical-microwave double resonance with an extremely narrow line width were observed in a rubidium atomic clock pumped by a semiconductor laser. It was confirmed that these novel spectral line shapes of double resonance were due to the modulation-transfer effect. These spectral line shapes were calculated by solving the equation of motion of the density matrix that describes the modulation-transfer process. The calculated results agree with the experimental results. By using the calculated results of the influence of the optical Stark effect on these spectral line shapes, characteristics of the inhomogeneous light shift were evaluated quantitatively for the frequency-modulated microwave. Furthermore, optimum values of modulation parameters were found by computer simulation in order to obtain the highest microwave frequency stability of the atomic clocks. An experiment was carried out employing these optimum parameters, and a short-term frequency stability as high as σy(τ) = 7.9 × 10−13τ−1/2 was obtained, where τ is an integration time of measurement.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. L. Picque, “Hyperfine optical pumping of a cesium atomic beam, and applications,” Metrologia 13, 115–119 (1977).
    [CrossRef]
  2. L. L. Lewis, M. Feldman, “Optical pumping by lasers in atomic frequency standards,” in Proceedings of the 35th Annual Frequency Control Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), p. 625.
  3. J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped rubidium atomic frequency standard,” J. Appl. Phys. 59, 3313–3317 (1986).
    [CrossRef]
  4. M. Hashimoto, M. Ohtsu, “Experiments on a semiconductor laser pumped Rb atomic clock,” IEEE J. Quantum Electron. QE-23, 446–451 (1987).
    [CrossRef]
  5. W. Happer, B. S. Mathur, “Effective operator formalism in optical pumping,” Phys. Rev. 163, 12–25 (1967).
    [CrossRef]
  6. J. C. Camparo, R. P. Frueholz, C. H. Volk, “Inhomogeneous light shift in a alkali-metal atoms,” Phys. Rev. 27, 1914–1925 (1983).
    [CrossRef]
  7. T. McClelland, L. K. Lam, T. M. Kwon, “Anomalous narrowing of magnetic-resonance linewidths in optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1967–1707 (1986).
    [CrossRef]
  8. J. Vanier, L. G. Bernier, “On the signal-to-noise ratio and short-term stability of passive rubidium frequency standards,” IEEE Trans. Instrum. Meas. IM-13, 221–230 (1981).
  9. T. W. Hänsch, “Nonlinear high-resolution spectroscopy of atoms and molecules,” in Nonlinear Spectroscopy, N. Bloembergen, ed. (North-Holland, Amsterdam, 1977), pp. 17–86.
  10. M. Ducloy, D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Phys. (Paris) 43, 57–65 (1982).
    [CrossRef]
  11. R. W. Boyd, M. Sargent, “Population pulsation and the dynamic Stark effect,” J. Opt. Soc. Am. B 5, 99–111 (1988).
    [CrossRef]
  12. M. T. Gruneisen, K. R. MacDonald, R. W. Boyd, “Induced gain and modified absorption of a weak probe in a strongly driven sodium vapor,” J. Opt. Soc. Am. B 5, 123–128 (1988).
    [CrossRef]
  13. J. H. Shirley, “Modulation transfer process in optical heterodyne saturation spectroscopy,” Opt. Lett. 7, 537–539 (1982).
    [CrossRef] [PubMed]
  14. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980).
    [CrossRef]
  15. B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11–19 (1968).
    [CrossRef]
  16. G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 51, 1379 (1973).
    [CrossRef]
  17. M. Ohtsu, “Realization of ultrahigh coherence in semiconductor lasers by negative electric feedback,” IEEE J. Lightwave Technol. 6, 245–256 (1988).
    [CrossRef]
  18. T. W. Hänsch, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
    [CrossRef]

1988

1987

M. Hashimoto, M. Ohtsu, “Experiments on a semiconductor laser pumped Rb atomic clock,” IEEE J. Quantum Electron. QE-23, 446–451 (1987).
[CrossRef]

1986

J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped rubidium atomic frequency standard,” J. Appl. Phys. 59, 3313–3317 (1986).
[CrossRef]

T. McClelland, L. K. Lam, T. M. Kwon, “Anomalous narrowing of magnetic-resonance linewidths in optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1967–1707 (1986).
[CrossRef]

1983

J. C. Camparo, R. P. Frueholz, C. H. Volk, “Inhomogeneous light shift in a alkali-metal atoms,” Phys. Rev. 27, 1914–1925 (1983).
[CrossRef]

1982

M. Ducloy, D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Phys. (Paris) 43, 57–65 (1982).
[CrossRef]

J. H. Shirley, “Modulation transfer process in optical heterodyne saturation spectroscopy,” Opt. Lett. 7, 537–539 (1982).
[CrossRef] [PubMed]

1981

J. Vanier, L. G. Bernier, “On the signal-to-noise ratio and short-term stability of passive rubidium frequency standards,” IEEE Trans. Instrum. Meas. IM-13, 221–230 (1981).

1980

1977

J. L. Picque, “Hyperfine optical pumping of a cesium atomic beam, and applications,” Metrologia 13, 115–119 (1977).
[CrossRef]

1973

G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 51, 1379 (1973).
[CrossRef]

1970

T. W. Hänsch, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
[CrossRef]

1968

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11–19 (1968).
[CrossRef]

1967

W. Happer, B. S. Mathur, “Effective operator formalism in optical pumping,” Phys. Rev. 163, 12–25 (1967).
[CrossRef]

Bernier, L. G.

J. Vanier, L. G. Bernier, “On the signal-to-noise ratio and short-term stability of passive rubidium frequency standards,” IEEE Trans. Instrum. Meas. IM-13, 221–230 (1981).

Bjorklund, G. C.

Bloch, D.

M. Ducloy, D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Phys. (Paris) 43, 57–65 (1982).
[CrossRef]

Boyd, R. W.

Busca, G.

G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 51, 1379 (1973).
[CrossRef]

Camparo, J. C.

J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped rubidium atomic frequency standard,” J. Appl. Phys. 59, 3313–3317 (1986).
[CrossRef]

J. C. Camparo, R. P. Frueholz, C. H. Volk, “Inhomogeneous light shift in a alkali-metal atoms,” Phys. Rev. 27, 1914–1925 (1983).
[CrossRef]

Ducloy, M.

M. Ducloy, D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Phys. (Paris) 43, 57–65 (1982).
[CrossRef]

Feldman, M.

L. L. Lewis, M. Feldman, “Optical pumping by lasers in atomic frequency standards,” in Proceedings of the 35th Annual Frequency Control Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), p. 625.

Frueholz, R. P.

J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped rubidium atomic frequency standard,” J. Appl. Phys. 59, 3313–3317 (1986).
[CrossRef]

J. C. Camparo, R. P. Frueholz, C. H. Volk, “Inhomogeneous light shift in a alkali-metal atoms,” Phys. Rev. 27, 1914–1925 (1983).
[CrossRef]

Gruneisen, M. T.

Hänsch, T. W.

T. W. Hänsch, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
[CrossRef]

T. W. Hänsch, “Nonlinear high-resolution spectroscopy of atoms and molecules,” in Nonlinear Spectroscopy, N. Bloembergen, ed. (North-Holland, Amsterdam, 1977), pp. 17–86.

Happer, W.

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11–19 (1968).
[CrossRef]

W. Happer, B. S. Mathur, “Effective operator formalism in optical pumping,” Phys. Rev. 163, 12–25 (1967).
[CrossRef]

Hashimoto, M.

M. Hashimoto, M. Ohtsu, “Experiments on a semiconductor laser pumped Rb atomic clock,” IEEE J. Quantum Electron. QE-23, 446–451 (1987).
[CrossRef]

Kwon, T. M.

T. McClelland, L. K. Lam, T. M. Kwon, “Anomalous narrowing of magnetic-resonance linewidths in optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1967–1707 (1986).
[CrossRef]

Lam, L. K.

T. McClelland, L. K. Lam, T. M. Kwon, “Anomalous narrowing of magnetic-resonance linewidths in optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1967–1707 (1986).
[CrossRef]

Lewis, L. L.

L. L. Lewis, M. Feldman, “Optical pumping by lasers in atomic frequency standards,” in Proceedings of the 35th Annual Frequency Control Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), p. 625.

MacDonald, K. R.

Mathur, B. S.

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11–19 (1968).
[CrossRef]

W. Happer, B. S. Mathur, “Effective operator formalism in optical pumping,” Phys. Rev. 163, 12–25 (1967).
[CrossRef]

McClelland, T.

T. McClelland, L. K. Lam, T. M. Kwon, “Anomalous narrowing of magnetic-resonance linewidths in optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1967–1707 (1986).
[CrossRef]

Ohtsu, M.

M. Ohtsu, “Realization of ultrahigh coherence in semiconductor lasers by negative electric feedback,” IEEE J. Lightwave Technol. 6, 245–256 (1988).
[CrossRef]

M. Hashimoto, M. Ohtsu, “Experiments on a semiconductor laser pumped Rb atomic clock,” IEEE J. Quantum Electron. QE-23, 446–451 (1987).
[CrossRef]

Picque, J. L.

J. L. Picque, “Hyperfine optical pumping of a cesium atomic beam, and applications,” Metrologia 13, 115–119 (1977).
[CrossRef]

Sargent, M.

Shirley, J. H.

Tang, H.

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11–19 (1968).
[CrossRef]

Tetu, M.

G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 51, 1379 (1973).
[CrossRef]

Vanier, J.

J. Vanier, L. G. Bernier, “On the signal-to-noise ratio and short-term stability of passive rubidium frequency standards,” IEEE Trans. Instrum. Meas. IM-13, 221–230 (1981).

G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 51, 1379 (1973).
[CrossRef]

Volk, C. H.

J. C. Camparo, R. P. Frueholz, C. H. Volk, “Inhomogeneous light shift in a alkali-metal atoms,” Phys. Rev. 27, 1914–1925 (1983).
[CrossRef]

Can. J. Phys.

G. Busca, M. Tetu, J. Vanier, “Light shift and light broadening in the 87Rb maser,” Can. J. Phys. 51, 1379 (1973).
[CrossRef]

IEEE J. Lightwave Technol.

M. Ohtsu, “Realization of ultrahigh coherence in semiconductor lasers by negative electric feedback,” IEEE J. Lightwave Technol. 6, 245–256 (1988).
[CrossRef]

IEEE J. Quantum Electron.

M. Hashimoto, M. Ohtsu, “Experiments on a semiconductor laser pumped Rb atomic clock,” IEEE J. Quantum Electron. QE-23, 446–451 (1987).
[CrossRef]

IEEE Trans. Instrum. Meas.

J. Vanier, L. G. Bernier, “On the signal-to-noise ratio and short-term stability of passive rubidium frequency standards,” IEEE Trans. Instrum. Meas. IM-13, 221–230 (1981).

J. Appl. Phys.

J. C. Camparo, R. P. Frueholz, “Fundamental stability limits for the diode-laser-pumped rubidium atomic frequency standard,” J. Appl. Phys. 59, 3313–3317 (1986).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. (Paris)

M. Ducloy, D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened media. Doppler-free heterodyne spectroscopy via collinear four-wave mixing in two- and three-level systems,” J. Phys. (Paris) 43, 57–65 (1982).
[CrossRef]

Metrologia

J. L. Picque, “Hyperfine optical pumping of a cesium atomic beam, and applications,” Metrologia 13, 115–119 (1977).
[CrossRef]

Opt. Lett.

Phys. Rev.

B. S. Mathur, H. Tang, W. Happer, “Light shifts in the alkali atoms,” Phys. Rev. 171, 11–19 (1968).
[CrossRef]

W. Happer, B. S. Mathur, “Effective operator formalism in optical pumping,” Phys. Rev. 163, 12–25 (1967).
[CrossRef]

J. C. Camparo, R. P. Frueholz, C. H. Volk, “Inhomogeneous light shift in a alkali-metal atoms,” Phys. Rev. 27, 1914–1925 (1983).
[CrossRef]

Phys. Rev. A

T. McClelland, L. K. Lam, T. M. Kwon, “Anomalous narrowing of magnetic-resonance linewidths in optically pumped alkali-metal vapor,” Phys. Rev. A 33, 1967–1707 (1986).
[CrossRef]

Z. Phys.

T. W. Hänsch, “Theory of a three-level gas laser amplifier,” Z. Phys. 236, 213–244 (1970).
[CrossRef]

Other

L. L. Lewis, M. Feldman, “Optical pumping by lasers in atomic frequency standards,” in Proceedings of the 35th Annual Frequency Control Symposium (Institute of Electrical and Electronics Engineers, New York, 1981), p. 625.

T. W. Hänsch, “Nonlinear high-resolution spectroscopy of atoms and molecules,” in Nonlinear Spectroscopy, N. Bloembergen, ed. (North-Holland, Amsterdam, 1977), pp. 17–86.

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 (14)

Fig. 1
Fig. 1

Block diagram of a semiconductor-laser-pumped 87Rb atomic clock: P.S.D., phase-sensitive detector; V.C.X.O., voltage-controlled crystal oscillator; A, postdetector amplifier; ωm, angular frequency for microwave frequency modulation.

Fig. 2
Fig. 2

Examples of novel line shapes of optical-microwave double resonance, (a) Modulation frequency, modulation index, and phase ωm/2π = 160 Hz, M = 1.8, and θ = 235°, respectively, (b) Modulation frequency, modulation index, and phase ωm/2π = 200 Hz, M = 1.6, and 0 = 270°, respectively. VPSD, Output voltage from the phase-sensitive detector. The laser-light intensity was I0 ≡ 14.4 μW/cm2. For definitions of ωM and ω12, see Appendix A.

Fig. 3
Fig. 3

Energy-level diagram of a 87Rb atom.

Fig. 4
Fig. 4

Measured dependences of Cq and Sq on microwave frequency, where the modulation frequency was ωm/2π = 1 kHz and the laser-light intensity was I0 = 74.4 μW/cm2. Modulation indices M were (a)–(c) 1.4 and (d) 4.3. For definitions of ωM and ω12, see Appendix A.

Fig. 5
Fig. 5

Calculated dependences of Cq and Sq on microwave frequency. Modulation indices M were (a)–(c) 1.4 and (d) 4.3. The modulation frequency was ωm/2π = 1 kHz, and the laser Rabi frequency was xL/2π = 400 kHz. For definitions of ωM and ω12, see Appendix A.

Fig. 6
Fig. 6

Calculated results of novel optical-microwave double-resonance line shapes, (a) Modulation frequency, modulation index, and phase ωm/2π = 160 Hz, M = 1.8, and θ = 235°, respectively, (b) Modulation frequency, modulation index, and phase ωm/2π = 200 Hz, M = 1.6, and θ = 270°, respectively. VPSD. Output voltage from the phase-sensitive detector. The laser Rabi frequency was xL/2π = 150 kHz. For definitions of ωM and ω12 see Appendix A.

Fig. 7
Fig. 7

Measured spectral line shapes of deformed S3. Laser-light intensities I0 were (a) 1440 μW/cm2, (b) 420 μW/cm2, and (c) 210 μW/cm2. The laser frequency detuning (ωLω13)/2π was A, 1200 MHz; B, 0 MHz, and C, −1200 MHz. The modulation index was M = 3.9. For definitions of ωL, ωM, ω13, and ω12, see Appendix A.

Fig. 8
Fig. 8

Measured characteristics of the light shift. The laser-light intensities I0 were A, 144 μW/cm2; B, 360 μW/cm2; and C, 1008 μW/ cm2. The modulation index was M = 3.9. ΔωLS, Shift of zero-crossing point of S3. For definitions of ωL, ω13, and ω12, see Appendix A.

Fig. 9
Fig. 9

Measured population inversion between the levels |1〉 and |2〉. The laser-light intensities I0 were A, 144 μW/cm2; B, 360 μW/ cm2; and C, 1008 μW/cm2. The modulation index was M = 2.4. C2, In-phase component of second harmonics. For definitions of ωL and ω13, see Appendix A.

Fig. 10
Fig. 10

Calculated line shapes of deformed S3. Laser Rabi frequencies xL/2π: (a) 1800 kHz, (b) 900 kHz, and (c) 500 kHz. The laser frequency detuning (ωLω13)/2π was A, 1200 MHz; B, 0 MHz; and C, −1200 MHz. The modulation index was M = 3.9. For definitions of ωL, ωM, ω13, and ω12, see Appendix A.

Fig. 11
Fig. 11

Calculated dependence of the frequency-discrimination sensitivity D of the fundamental component on the normalized modulation frequency ωm/γ21′ and the phase difference θ, where the modulation index M was fixed as 1.2.

Fig. 12
Fig. 12

Calculated dependence of the frequency-discrimination sensitivity D of the fundamental component on the normalized modulation frequency ωm21′, where the phase difference θ was fixed at 120°. The modulation indices M were A, 0.8; B, 1.2; C, 1.6; and D, 1.8.

Fig. 13
Fig. 13

Calculated dependence of the frequency discrimination sensitivity D of the third-harmonic component on the normalized modulation frequency ωm21′ and the phase difference θ, where the modulation index M was fixed at 3.6.

Fig. 14
Fig. 14

Calculated dependence of the frequency-discrimination sensitivity D of the third-harmonic component on the normalized modulation frequency ωm21′, where the phase difference θ was fixed at 270°. The modulation indices M were A, 3.4; B, 3.6; C, 3.8; and D, 4.0.

Tables (2)

Tables Icon

Table 1 Constants Used in Our Calculationsa

Tables Icon

Table 2 Measured Values of S/N and Linewidth (Δν)

Equations (71)

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

σ y ( τ ) = 0.11 Q ( S / N ) τ 1 / 2 ,
Q = ω 12 / 2 π Δ ν ,
σ y ( τ ) = 0.11 N ( ω 12 / 2 π ) D τ 1 / 2 ,
H M ( r , z , t ) = q = J q ( M ) H M cos [ ( ω M + q ω m ) t ] ,
H M = H M 0 J 0 ( 3.832 r / R ) sin [ ( k M + q k m ) z ] .
E L ( z , t ) = E L cos ( ω L t k L z ) ,
I DET ( t ) = c 0 | E L ( z , t ) exp [ α L c ] | 2 c 0 E L 2 ( 1 α L c α * L c ) ,
d ρ 11 d t = i ( μ 21 H M ρ 21 * + μ 31 E L ρ 13 μ 21 * H M * ρ 21 μ 31 * E L * ρ 13 * ) ( ρ 11 n 1 ) γ 1 ,
d ρ 22 d t = i ( μ 21 H M ρ 21 * μ 21 * H M * ρ 21 ) ( ρ 22 n 2 ) γ 2 ,
d ρ 33 d t = i ( μ 31 E L ρ 13 μ 31 * E L * ρ 13 * ) ( ρ 33 n 3 ) γ 3 ,
d ρ 21 d t = i [ ( ρ 22 ρ 11 ) μ 21 H M + μ 31 E L ρ 32 * ] + i ( ω 12 + i γ 21 ) ρ 21 ,
d ρ 32 d t = i ( μ 21 * H M * ρ 13 * μ 31 E L ρ 21 * ) + i ( ω 23 + i γ 23 ) ρ 32 ,
d ρ 13 d t = i [ ( ρ 33 ρ 11 ) μ 31 * E L * + μ 21 * H M * ρ 32 * ] i ( ω 13 i γ 13 ) ρ 13 ,
α ( t ) = i k L μ 31 2 E L * p 13 = k L μ 31 2 2 { i ( ρ 33 ρ 11 ) Δ 13 + x M 2 4 × q p i J p + q J p ( ρ 22 ρ 11 ) Δ 13 [ Δ 21 ( p ) Δ 32 * ( p ) + x L 2 / 4 ] exp ( i q ω m t ) } ,
I DET ( t ) = I 0 [ 1 2 δ OPT q p J p + q J p ( δ p i ϕ p ) exp ( i q ω m t ) q p J p + q J p ( δ p + i ϕ p ) exp ( i q ω m t ) ] ,
I 0 = c 0 E L 2 ,
δ OPT = k L μ 31 2 L c 2 ( ρ 33 ρ 11 ) γ 13 ( ω 13 ω L ) 2 + γ 13 2 ,
δ p = k L μ 31 2 L c 2 x M 2 4 Re { i ( ρ 22 ρ 11 ) Δ 13 [ Δ 21 ( p ) Δ 32 * ( p ) + x L 2 / 4 ] } ,
ϕ p = k L μ 31 2 L c 2 x M 2 4 Im { i ( ρ 22 ρ 11 ) Δ 13 [ Δ 21 ( p ) Δ 32 * ( p ) + x L 2 / 4 ] } ,
I DET ( t ) = I 0 ( 1 2 δ OPT ) + q = 0 [ C q cos ( q ω m t ) + S q sin ( q ω m t ) ] ,
C q = I 0 p = ( J p + q + J p q ) J p δ p
S q = I 0 p = ( J p + q J p q ) J p ϕ p .
C 0 = I 0 [ 2 J 0 2 δ 0 J 1 2 ( δ 1 + δ 1 ) J 2 2 ( δ 2 + δ 2 ) J 3 2 ( δ 3 + δ 3 ) ] ,
C 1 = I 0 [ J 0 J 1 ( δ 1 δ 1 ) J 1 J 2 ( δ 2 + δ 1 δ 1 δ 2 ) J 2 J 3 ( δ 3 + δ 2 δ 2 δ 3 ) J 3 J 4 ( δ 3 δ 3 ) ] ,
S 1 = I 0 [ J 0 J 1 ( ϕ 1 2 ϕ 0 + ϕ 1 ) + J 1 J 2 ( ϕ 2 ϕ 1 ϕ 1 + ϕ 2 ) + J 2 J 3 ( ϕ 3 ϕ 2 ϕ 2 + ϕ 3 ) J 3 J 4 ( ϕ 3 + ϕ 3 ) ] ,
C 2 = I 0 [ J 0 J 2 ( δ 2 + 2 δ 0 + δ 2 ) + J 1 2 ( δ 1 + δ 1 ) J 1 J 3 ( δ 3 + δ 1 + δ 1 + δ 3 ) J 2 J 4 ( δ 2 + δ 2 ) J 3 J 5 ( δ 3 + δ 3 ) ] ,
S 2 = I 0 [ J 0 J 2 ( ϕ 2 ϕ 2 ) J 1 2 ( ϕ 1 ϕ 1 ) + J 1 J 3 ( ϕ 3 ϕ 1 + ϕ 1 ϕ 3 ) J 2 J 4 ( ϕ 2 ϕ 2 ) J 3 J 5 ( ϕ 3 ϕ 3 ) ] ,
C 3 = I 0 [ J 0 J 3 ( δ 3 δ 3 ) + J 1 J 2 ( δ 2 δ 1 + δ 1 δ 2 ) J 1 J 4 ( δ 1 δ 1 ) J 2 J 5 ( δ 2 δ 2 ) J 3 J 6 ( δ 3 δ 3 ) ] ,
S 3 = I 0 [ J 0 J 3 ( ϕ 3 2 ϕ 0 + ϕ 3 ) J 1 J 2 ( ϕ 2 ϕ 1 ϕ 1 + ϕ 2 ) J 1 J 4 ( ϕ 1 + ϕ 1 ) J 2 J 5 ( ϕ 2 + ϕ 2 ) J 3 J 6 ( ϕ 3 + ϕ 3 ) ] .
C 1 I 0 J 0 J 1 ( δ 1 δ 1 ) = I 0 J 0 J 1 [ δ ( ω M + ω m ) δ ( ω M ω m ) ] I 0 J 0 J 1 d δ ( ω M ) d ω M
S 1 I 0 J 0 J 1 ( ϕ 1 2 ϕ 0 + ϕ 1 ) = I 0 J 0 J 1 [ ϕ ( ω M + ω m ) 2 ϕ ( ω M ) + ϕ ( ω M ω m ) ] I 0 J 0 J 1 d 2 ϕ ( ω M ) d ω M 2 .
V PSD = V 0 ( C q 2 cos θ + S q 2 sin θ ) ,
Δ 13 [ Δ 21 ( p ) Δ 32 * ( p ) + x L 2 / 4 ] = Δ 13 [ ( ω 12 ω M p ω m + i γ 21 ) × ( ω 23 ω L + ω M + p ω m i γ 32 ) + x L 2 / 4 ] = Δ 13 [ ( ω 12 ω M p ω m + i γ 21 ) × ( ω 13 ω 12 ω L + ω M + p ω m i γ 32 + i γ 21 i γ 21 ) + x L 2 / 4 ] = Δ 13 [ Δ 21 ( p ) ( + ) ] [ Δ 21 ( p ) ( ) ] ,
( ± ) = ω 13 ω L i ( γ 32 γ 21 ) ± Ω 13 2
Ω 13 = [ { ω 13 ω L i ( γ 32 γ 21 ) } 2 + x L 2 ] 1 / 2 .
{ } = i ( ρ 22 ρ 11 ) Δ 13 Ω 13 [ 1 Δ 21 ( p ) ( ) 1 Δ 21 ( p ) ( + ) ] .
Ω 13 = [ ω 13 ω L i ( γ 32 γ 21 ) ] × [ 1 + x L 2 { ω 13 ω L i ( γ 32 γ 21 ) } 2 ] 1 / 2 ω 13 ω L i ( γ 32 γ 21 ) + x L 2 / 2 ω 13 ω L i ( γ 32 γ 21 ) .
( + ) = ω 13 ω L i γ 32 + x L 2 / 4 ω 13 ω L i ( γ 32 γ 21 )
( ) = x L 2 / 4 ω 13 ω L i ( γ 32 γ 21 ) .
{ } = i ( ρ 22 ρ 11 ) Δ 13 Ω 13 [ 1 ω 12 ω M p ω m + i γ 21 1 ω 12 ω M p ω m + i γ 21 ] ,
ω 12 = ω 12 + ω 13 ω L ( ω 13 ω L ) 2 + ( γ 32 γ 21 ) 2 x L 2 4 ,
γ 21 = γ 21 + γ 32 ( ω 13 ω L ) 2 + ( γ 32 γ 21 ) 2 x L 2 4 ·
ω 12 = ω 12 ω 13 + ω L ω 13 ω L ( ω 13 ω L ) 2 + ( γ 32 γ 21 ) 2 x L 2 4 ,
γ 21 = γ 32 γ 32 ( ω 13 ω L ) 2 + ( γ 32 γ 21 ) 2 x L 2 4 ·
{ } = ( ρ 22 ρ 11 ) | Δ 13 | | Ω 13 | exp ( i ζ ) i ( ω 12 ω M p ω m ) + γ 21 ( ω 12 ω M p ω m ) 2 + γ 21 2 ,
ζ tan 1 ω 13 ω L γ 13 tan 1 ω 13 ω L + ω 13 ω L ( ω 13 ω L ) 2 + ( γ 32 γ 21 ) 2 x L 2 2 γ 32 γ 32 ( ω 13 ω L ) 2 + ( γ 32 γ 21 ) 2 x L 2 .
x L 2 ( r ) = x L 2 ( 0 ) 2 π w exp [ ( r / w ) 2 ] ,
δ p = 2 π 0 R c k L μ 31 2 L c 2 x M 2 4 ( ρ 22 ρ 11 ) | Δ 13 | | Ω 13 | × γ 21 ( ω 12 ω M p ω m ) 2 + γ 21 2 2 π w exp [ ( r / w ) 2 ] d r
ϕ p = 2 π 0 R c k L μ 31 2 L c 2 x M 2 4 ( ρ 22 ρ 11 ) | Δ 13 | | Ω 13 | × ω 12 ω M p ω m ( ω 12 ω M p ω m ) 2 + γ 21 2 2 π w exp [ ( r / w ) 2 ] d r ,
D = 2 π d V PSD d ω M | ω M = ω 12 ,
δ p = γ 21 ( ω 12 ω M p ω m ) 2 + γ 21 2
ϕ p = ω 12 ω M p ω m ( ω 12 ω M p ω m ) 2 + γ 21 2 .
D MAX = 1.3 × 10 4 ( 1 / Hz ) at M = 1.2 , ω m / γ 21 = 1.0 , and θ = 120 ° .
D MAX = 6.1 × 10 5 ( 1 / Hz ) at M = 3.6 , ω m / γ 21 = 0.5 , and θ = 270 ° .
σ y ( τ ) = 7.9 × 10 13 τ 1 / 2
σ y ( τ ) = 5.0 × 10 12 τ 1 / 2 ,
D MAX = 1.3 × 10 4 ( 1 / Hz ) at M = 1.2 , ω m / γ 21 = 1.0 , and θ = 120 ° .
D MAX = 6.1 × 10 5 ( 1 / Hz ) at M = 3.6 , ω m / γ 21 = 0.5 , and θ = 270 ° .
ρ 21 = q p 21 ( q ) 2 exp [ i ( ω M + q ω m ) t ] ,
ρ 32 = q p 32 ( q ) 2 exp { i [ ( ω L ω M + q ω m ) t k L z ] } ,
ρ 13 = q p 13 ( q ) 2 exp { i [ ( ω L + q ω m ) t k L z ] } ,
p 21 ( q ) = 1 Δ 21 ( q ) [ ( ρ 22 ρ 11 ) x M + x L 2 p 32 * ( q ) ] ,
p 32 ( q ) = 1 Δ 32 ( q ) [ x M * 2 p J p + q p 13 * ( q ) x L 2 p 21 * ( q ) ] ,
p 13 ( q ) = 1 Δ 13 ( 0 ) [ ( ρ 33 ρ 11 ) x L * x M * 2 p J p p 32 * ( p ) ] ( q = 0 ) = 1 Δ 13 ( q ) [ x M * 2 p J p + q p 32 * ( p ) ] ( q 0 ) ,
x M = μ 21 H M ,
x L = μ 31 E L ,
Δ 21 ( q ) = ( ω 12 ω M q ω m ) + i γ 21 ,
Δ 32 ( q ) = ( ω 23 ω L + ω M + q ω m ) + i γ 32 ,
Δ 13 ( q ) = ( ω 13 ω L q ω m ) + i γ 13 ,
p 32 ( q ) = ( ρ 21 ρ 11 ) J q Δ 21 * ( q ) Δ 32 ( q ) + x L 2 / 4 x M * x L 2 .
ρ 13 = q p 13 ( q ) exp ( i q ω m t ) = { ( ρ 33 ρ 11 ) Δ 13 + x M 2 4 × q p J p + q J p ( ρ 33 ρ 11 ) Δ 13 [ Δ 21 ( p ) Δ 32 * ( p ) + x L 2 / 4 ] × exp ( i q ω m t ) } x L * } ,

Metrics