Abstract

We conducted a theoretical and experimental study of lineshape in modulation transfer spectroscopy for 87Rb atoms. When a linearly polarized pump beam, modulated at an angular frequency of Ω, overlaps in parallel with an unmodulated linearly polarized probe beam, combined modulated probe beams are generated via nonlinear interaction with atoms. The detected modulation transfer signals are calculated by numerically solving the complete optical Bloch equations for the 87Rb atoms without the use of any phenomenological parameters. We find that the calculated results are in good agreement with experimental results.

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References

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  1. W. Demtröder, Laser Spectroscopy (Springer, Berlin, 1998).
  2. C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36(20), 1170–1173 (1976).
    [CrossRef]
  3. K. L. Corwin, Z. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, “Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor,” Appl. Opt. 37(15), 3295–3298 (1998).
    [CrossRef]
  4. M. L. Harris, S. L. Cornish, A. Tripathi, and I. G. Hughes, “Optimization of sub-Doppler DAVLL on the rubidium D2 line,” J. Phys. B,  41(8), 085401 (2008).
    [CrossRef]
  5. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5(1), 15–17 (1980).
    [CrossRef] [PubMed]
  6. J. H. Shirley, “Modulation transfer processes in optical heterodyne saturation spectroscopy,” Opt. Lett. 7(11), 537–539 (1982).
    [CrossRef] [PubMed]
  7. D. J. McCarron, S. A. King, and S. L. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Meas. Sci. Technol. 19(10), 105601 (2008).
    [CrossRef]
  8. M. L. Eickhoff and J. L. Hall, “Optical frequency standard at 532 nm,” IEEE Trans. Instrum. Meas. 44(2), 155–158 (1995).
    [CrossRef]
  9. E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
    [CrossRef]
  10. A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25(5), 2606–2621 (1982).
    [CrossRef]
  11. E. Jaatinen, “Theoretical determination of maximum signal levels obtainable with modulation transfer spectroscopy,” Opt. Commun. 120(1–2), 91–97 (1995).
    [CrossRef]
  12. F. Bertinetto, P. Cordiale, G. Galzerano, and E. Bava, “Frequency stabilization of DBR diode laser against Cs absorption lines at 852 nm using the modulation transfer method,” IEEE Trans. Instrum. Meas. 50(2), 490–492 (2001).
    [CrossRef]
  13. J. Zhang, D. Wei, C. Xie, and K. Peng, “Characteristics of absorption and dispersion for rubidium D2 lines with the modulation transfer spectrum,” Opt. Express,  11(11), 1338–1344 (2003).
    [CrossRef] [PubMed]
  14. L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
    [CrossRef]
  15. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1960).
  16. J. Sagle, R. K. Namiotka, and J. Huennekens, “Measurement and modelling of intensity dependent absorption and transit relaxation on the cesium D1 line,” J. Phys. B 29(12), 2629–2643 (1996).
    [CrossRef]
  17. P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B 41(15), 155004 (2008).
    [CrossRef]

2011

L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
[CrossRef]

2008

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B 41(15), 155004 (2008).
[CrossRef]

M. L. Harris, S. L. Cornish, A. Tripathi, and I. G. Hughes, “Optimization of sub-Doppler DAVLL on the rubidium D2 line,” J. Phys. B,  41(8), 085401 (2008).
[CrossRef]

D. J. McCarron, S. A. King, and S. L. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Meas. Sci. Technol. 19(10), 105601 (2008).
[CrossRef]

2007

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

2003

2001

F. Bertinetto, P. Cordiale, G. Galzerano, and E. Bava, “Frequency stabilization of DBR diode laser against Cs absorption lines at 852 nm using the modulation transfer method,” IEEE Trans. Instrum. Meas. 50(2), 490–492 (2001).
[CrossRef]

1998

1996

J. Sagle, R. K. Namiotka, and J. Huennekens, “Measurement and modelling of intensity dependent absorption and transit relaxation on the cesium D1 line,” J. Phys. B 29(12), 2629–2643 (1996).
[CrossRef]

1995

E. Jaatinen, “Theoretical determination of maximum signal levels obtainable with modulation transfer spectroscopy,” Opt. Commun. 120(1–2), 91–97 (1995).
[CrossRef]

M. L. Eickhoff and J. L. Hall, “Optical frequency standard at 532 nm,” IEEE Trans. Instrum. Meas. 44(2), 155–158 (1995).
[CrossRef]

1982

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25(5), 2606–2621 (1982).
[CrossRef]

J. H. Shirley, “Modulation transfer processes in optical heterodyne saturation spectroscopy,” Opt. Lett. 7(11), 537–539 (1982).
[CrossRef] [PubMed]

1980

1976

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36(20), 1170–1173 (1976).
[CrossRef]

Adams, C. S.

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B 41(15), 155004 (2008).
[CrossRef]

Bava, E.

F. Bertinetto, P. Cordiale, G. Galzerano, and E. Bava, “Frequency stabilization of DBR diode laser against Cs absorption lines at 852 nm using the modulation transfer method,” IEEE Trans. Instrum. Meas. 50(2), 490–492 (2001).
[CrossRef]

Bertinetto, F.

F. Bertinetto, P. Cordiale, G. Galzerano, and E. Bava, “Frequency stabilization of DBR diode laser against Cs absorption lines at 852 nm using the modulation transfer method,” IEEE Trans. Instrum. Meas. 50(2), 490–492 (2001).
[CrossRef]

Bjorklund, G. C.

Brewer, R. G.

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25(5), 2606–2621 (1982).
[CrossRef]

Cho, C. H.

L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
[CrossRef]

Cho, H.

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

Cordiale, P.

F. Bertinetto, P. Cordiale, G. Galzerano, and E. Bava, “Frequency stabilization of DBR diode laser against Cs absorption lines at 852 nm using the modulation transfer method,” IEEE Trans. Instrum. Meas. 50(2), 490–492 (2001).
[CrossRef]

Cornish, S. L.

D. J. McCarron, S. A. King, and S. L. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Meas. Sci. Technol. 19(10), 105601 (2008).
[CrossRef]

M. L. Harris, S. L. Cornish, A. Tripathi, and I. G. Hughes, “Optimization of sub-Doppler DAVLL on the rubidium D2 line,” J. Phys. B,  41(8), 085401 (2008).
[CrossRef]

Corwin, K. L.

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer, Berlin, 1998).

DeVoe, R. G.

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25(5), 2606–2621 (1982).
[CrossRef]

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1960).

Eickhoff, M. L.

M. L. Eickhoff and J. L. Hall, “Optical frequency standard at 532 nm,” IEEE Trans. Instrum. Meas. 44(2), 155–158 (1995).
[CrossRef]

Epstein, R. J.

Galzerano, G.

F. Bertinetto, P. Cordiale, G. Galzerano, and E. Bava, “Frequency stabilization of DBR diode laser against Cs absorption lines at 852 nm using the modulation transfer method,” IEEE Trans. Instrum. Meas. 50(2), 490–492 (2001).
[CrossRef]

Ge, C.

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B 41(15), 155004 (2008).
[CrossRef]

Hall, J. L.

M. L. Eickhoff and J. L. Hall, “Optical frequency standard at 532 nm,” IEEE Trans. Instrum. Meas. 44(2), 155–158 (1995).
[CrossRef]

Hand, C. F.

Hänsch, T. W.

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36(20), 1170–1173 (1976).
[CrossRef]

Harris, M. L.

M. L. Harris, S. L. Cornish, A. Tripathi, and I. G. Hughes, “Optimization of sub-Doppler DAVLL on the rubidium D2 line,” J. Phys. B,  41(8), 085401 (2008).
[CrossRef]

Huennekens, J.

J. Sagle, R. K. Namiotka, and J. Huennekens, “Measurement and modelling of intensity dependent absorption and transit relaxation on the cesium D1 line,” J. Phys. B 29(12), 2629–2643 (1996).
[CrossRef]

Hughes, I. G.

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B 41(15), 155004 (2008).
[CrossRef]

M. L. Harris, S. L. Cornish, A. Tripathi, and I. G. Hughes, “Optimization of sub-Doppler DAVLL on the rubidium D2 line,” J. Phys. B,  41(8), 085401 (2008).
[CrossRef]

Jaatinen, E.

E. Jaatinen, “Theoretical determination of maximum signal levels obtainable with modulation transfer spectroscopy,” Opt. Commun. 120(1–2), 91–97 (1995).
[CrossRef]

Kim, E. B.

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

King, S. A.

D. J. McCarron, S. A. King, and S. L. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Meas. Sci. Technol. 19(10), 105601 (2008).
[CrossRef]

Kwon, T. Y.

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

Lee, H. S.

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

Li, L. Z.

L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
[CrossRef]

Lu, Z.

McCarron, D. J.

D. J. McCarron, S. A. King, and S. L. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Meas. Sci. Technol. 19(10), 105601 (2008).
[CrossRef]

Namiotka, R. K.

J. Sagle, R. K. Namiotka, and J. Huennekens, “Measurement and modelling of intensity dependent absorption and transit relaxation on the cesium D1 line,” J. Phys. B 29(12), 2629–2643 (1996).
[CrossRef]

Noh, H. R.

L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
[CrossRef]

Park, C. Y.

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

Park, J. D.

L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
[CrossRef]

Park, S. E.

L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
[CrossRef]

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

Park, Y. H.

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

Peng, K.

Sagle, J.

J. Sagle, R. K. Namiotka, and J. Huennekens, “Measurement and modelling of intensity dependent absorption and transit relaxation on the cesium D1 line,” J. Phys. B 29(12), 2629–2643 (1996).
[CrossRef]

Schenzle, A.

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25(5), 2606–2621 (1982).
[CrossRef]

Shirley, J. H.

Siddons, P.

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B 41(15), 155004 (2008).
[CrossRef]

Tripathi, A.

M. L. Harris, S. L. Cornish, A. Tripathi, and I. G. Hughes, “Optimization of sub-Doppler DAVLL on the rubidium D2 line,” J. Phys. B,  41(8), 085401 (2008).
[CrossRef]

Wei, D.

Wieman, C.

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36(20), 1170–1173 (1976).
[CrossRef]

Wieman, C. E.

Xie, C.

Yee, D.-S.

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

Zhang, J.

Appl. Opt.

IEEE Trans. Instrum. Meas.

M. L. Eickhoff and J. L. Hall, “Optical frequency standard at 532 nm,” IEEE Trans. Instrum. Meas. 44(2), 155–158 (1995).
[CrossRef]

E. B. Kim, S. E. Park, C. Y. Park, Y. H. Park, D.-S. Yee, T. Y. Kwon, H. S. Lee, and H. Cho, “Absolute frequency measurement of F = 4 → F′ = 5 transition line of cesium using amplified optical frequency comb,” IEEE Trans. Instrum. Meas. 56(2), 448–452 (2007).
[CrossRef]

F. Bertinetto, P. Cordiale, G. Galzerano, and E. Bava, “Frequency stabilization of DBR diode laser against Cs absorption lines at 852 nm using the modulation transfer method,” IEEE Trans. Instrum. Meas. 50(2), 490–492 (2001).
[CrossRef]

J. Phys. B

J. Sagle, R. K. Namiotka, and J. Huennekens, “Measurement and modelling of intensity dependent absorption and transit relaxation on the cesium D1 line,” J. Phys. B 29(12), 2629–2643 (1996).
[CrossRef]

P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium D lines: comparison between theory and experiment,” J. Phys. B 41(15), 155004 (2008).
[CrossRef]

M. L. Harris, S. L. Cornish, A. Tripathi, and I. G. Hughes, “Optimization of sub-Doppler DAVLL on the rubidium D2 line,” J. Phys. B,  41(8), 085401 (2008).
[CrossRef]

J. Phys. Soc. Jpn.

L. Z. Li, S. E. Park, H. R. Noh, J. D. Park, and C. H. Cho, “Modulation transfer spectroscopy for a two-level atomic system with a non-cycling transition,” J. Phys. Soc. Jpn. 80(7), 074301 (2011).
[CrossRef]

Meas. Sci. Technol.

D. J. McCarron, S. A. King, and S. L. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Meas. Sci. Technol. 19(10), 105601 (2008).
[CrossRef]

Opt. Commun.

E. Jaatinen, “Theoretical determination of maximum signal levels obtainable with modulation transfer spectroscopy,” Opt. Commun. 120(1–2), 91–97 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

A. Schenzle, R. G. DeVoe, and R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25(5), 2606–2621 (1982).
[CrossRef]

Phys. Rev. Lett.

C. Wieman and T. W. Hänsch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36(20), 1170–1173 (1976).
[CrossRef]

Other

W. Demtröder, Laser Spectroscopy (Springer, Berlin, 1998).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1960).

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic diagram of modulation transfer spectroscopy. (b) Energy level diagram of 87Rb atom.

Fig. 2
Fig. 2

The MTS signals for two-level atoms at the modulation frequency of (a) Ω/(2π) = 30 MHz and (b) Ω/(2π) = 3 MHz.

Fig. 3
Fig. 3

(a) The calculated and (b) experimental results of the MTS spectra for the transitions Fg = 2 → Fe = 1, 2, 3 of 87Rb atoms.

Fig. 4
Fig. 4

(a) The calculated and (b) experimental results of the MTS spectra for the transitions Fg = 1 → Fe = 0, 1, 2 of 87Rb atoms.

Fig. 5
Fig. 5

Energy level diagrams for the transitions Fg = 1 → Fe = 0 where the linearly polarized carrier (c) and sideband (s) beams are (a) in parallel and (b) perpendicular to the linearly polarized probe beam (p). (c) Diagram for the crossover signal ( CO 0 1).

Equations (13)

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

ρ ˙ = ( i / h ¯ ) [ H 0 + V , ρ ] + ρ ˙ sp ,
H 0 = F e = F g 1 F g + 1 m e = F e F e h ¯ ( ω 0 Δ F e F g + 1 ) | F e , m e F e , m e | ,
V = F e = F g 1 F g m = F g F g h ¯ 2 C F g , m F e , m ( Ω c e i ω 1 t + Ω s e i ( ω 1 + Ω ) t Ω s e i ( ω 1 Ω ) t + Ω p e i ω 2 t ) | F e , m F g , m | + h . c . ,
C F g , m g F e , m e = ( 1 ) 2 F e + I + J g + J e + L g + S + m F g + 1 × ( 2 L e + 1 ) ( 2 J e + 1 ) ( 2 J g + 1 ) ( 2 F e + 1 ) ( 2 F g + 1 ) × { L g L e 1 J e J g S } { J g J e 1 F e F g I } ( F e 1 F g m e m g m e m g ) ,
F e , m | ρ ˙ sp | F e , m = Γ F e , m | ρ | F e , m , F e , m | ρ ˙ sp | F g , m = ( Γ / 2 ) F e , m | ρ | F g , m , F g , m | ρ ˙ sp | F g , m = Γ F e = F g 1 F g + 1 q = 1 1 C F g , m F e , m + q C F g , m F e , m + q F g , m | ρ | F g , m ,
σ ˙ i j = e i c i j t ρ ˙ i j i c i j σ i j ,
σ j j = p j j 1 + ( p j j 2 + i p j j 3 ) e i Ω t + ( p j j 2 i p j j 3 ) e i Ω t ,
σ i j = ( r i j 1 + i s i j 1 ) e i ( δ p Ω ) t + ( r i j 2 + i s i j 2 ) e i ( δ p + Ω ) t + ( r i j 3 + i s i j 3 ) e i δ p t + other 5 terms ,
d = Tr ( ρ d ) = d e g i j C i j { ( r i j 1 + i s i j 1 ) e i ( ω 2 Ω ) t + ( r i j 2 + i s i j 2 ) e i ( ω 2 + Ω ) t + ( r i j 3 + i s i j 3 ) e i ω 2 t + other 5 terms + c . c } ,
I 0 ( δ 1 , δ p , t ) cos Ω t + Q 0 ( δ 1 , δ p , t ) sin Ω t ,
I 0 ( δ 1 , δ p , t ) = i j C i j ( s i j 2 + s i j 1 ) , Q 0 ( δ 1 , δ p , t ) = i j C i j ( r i j 2 r i j 1 ) ,
I = 1 t av 0 t av d t d v f D ( v ) I 0 + ( δ + k v , 2 k v , t ) Q = 1 t av 0 t av d t d v f D ( v ) Q 0 ( δ + k v , 2 k v , t ) ,
f D = ( π u ) 1 exp [ ( v / u ) 2 ]

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