Abstract

We demonstrate a technique for imaging magnetic fields using velocity-selective two-photon resonances in a cold atom cloud. Freely expanding 85Rb atoms released from a magneto-optical trap are exposed to a brief (≈1 ms), off-resonant, retro-reflected laser pulse in a lin-perp-lin configuration. Two-photon resonance between magnetic sublevels occurs only for atoms in narrow velocity classes dependent on the magnetic field strength. The momentum of resonant atoms is altered by the pulse, and this two-photon momentum change is easily visible after further ballistic expansion. When the momentum pulse is applied to an atom cloud with finite size, magnetic field variations across the sample result in position-dependent features in images of the expanded cloud. We demonstrate the technique by imaging magnetic field variations over ≈5 mm with ≈250µm spatial resolution.

© 2008 Optical Society of America

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  1. M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
    [CrossRef] [PubMed]
  2. V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004).
    [CrossRef]
  3. J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
    [CrossRef]
  4. J. Chab??e, H. Lignier, P. Szriftgiser, and J. C. Garreau, "Improving Raman velocimetry of laser-cooled cesium atoms by spin-polarization," Opt. Commun. 274, 254-259 (2007).
    [CrossRef]
  5. D. D. McGregor, "High-sensitivity helium resonance magnetometers," Review of Scientific Instruments 58(6), 1067-1076 (1987). URL http://link.aip.org/link/?RSI/58/1067/1.
    [CrossRef]
  6. D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
    [CrossRef]
  7. T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
    [CrossRef]
  8. M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, "Faraday spectroscopy in a dark optical trap," Phys. Rev. A 77, 063417 (2008).
    [CrossRef]
  9. M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
    [CrossRef] [PubMed]
  10. M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
    [CrossRef]
  11. E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004).
    [CrossRef]

2008 (1)

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, "Faraday spectroscopy in a dark optical trap," Phys. Rev. A 77, 063417 (2008).
[CrossRef]

2007 (3)

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
[CrossRef]

J. Chab??e, H. Lignier, P. Szriftgiser, and J. C. Garreau, "Improving Raman velocimetry of laser-cooled cesium atoms by spin-polarization," Opt. Commun. 274, 254-259 (2007).
[CrossRef]

2004 (2)

V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004).
[CrossRef]

E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004).
[CrossRef]

2002 (1)

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
[CrossRef]

2000 (1)

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
[CrossRef]

1999 (1)

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

1991 (1)

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

Alexandrov, E. B.

E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004).
[CrossRef]

Balabas, M. V.

E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004).
[CrossRef]

Bashkansky, M.

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, "Faraday spectroscopy in a dark optical trap," Phys. Rev. A 77, 063417 (2008).
[CrossRef]

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
[CrossRef]

Boyer, V.

V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004).
[CrossRef]

Budker, D.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
[CrossRef]

Chu, S.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

Dutton, Z.

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
[CrossRef]

Fatemi, F. K.

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, "Faraday spectroscopy in a dark optical trap," Phys. Rev. A 77, 063417 (2008).
[CrossRef]

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
[CrossRef]

Fixler, J. B.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
[CrossRef]

Foster, G. T.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
[CrossRef]

Guzman, J.

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

Higbie, J. M.

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

Isayama, T.

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

Ishikawa, K.

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

Kasapi, S.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

Kasevich, M.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

Kasevich, M. A.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
[CrossRef]

Kimball, D. F.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
[CrossRef]

Leslie, S. R.

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

Lising, L. J.

V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004).
[CrossRef]

McGuirk, J. M.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
[CrossRef]

Moler, K.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

Olson, S. E.

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
[CrossRef]

Pazgalev, A. S.

E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004).
[CrossRef]

Phillips, W. D.

V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004).
[CrossRef]

Riis, E.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

Rochester, S. M.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
[CrossRef]

Rolston, S. L.

V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004).
[CrossRef]

Sadler, L. E.

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

Snadden, M. J.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
[CrossRef]

Stamper-Kurn, D. M.

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

Takahashi, Y.

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

Tanaka, N.

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

Terraciano, M. L.

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, "Faraday spectroscopy in a dark optical trap," Phys. Rev. A 77, 063417 (2008).
[CrossRef]

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
[CrossRef]

Toyoda, K.

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

Vengalattore, M.

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

Vershovski, A. K.

E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004).
[CrossRef]

Weiss, D. S.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

Yabuzaki, T.

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

Yashchuk, V. V.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
[CrossRef]

Zolotorev, M.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
[CrossRef]

Opt. Commun. (1)

J. Chab??e, H. Lignier, P. Szriftgiser, and J. C. Garreau, "Improving Raman velocimetry of laser-cooled cesium atoms by spin-polarization," Opt. Commun. 274, 254-259 (2007).
[CrossRef]

Phys. Rev. A (6)

V. Boyer, L. J. Lising, S. L. Rolston, and W. D. Phillips, "Deeply subrecoil two-dimensional Raman cooling," Phys. Rev. A 70(4), 043405 (pages 8) (2004).
[CrossRef]

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, "Sensitive absolute-gravity gradiometry using atom interferometry," Phys. Rev. A 65(3), 033608 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and M. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62(4), 043403 (2000).
[CrossRef]

T. Isayama, Y. Takahashi, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, "Observation of Larmor spin precession of laser-cooled Rb atoms via paramagnetic Faraday rotation," Phys. Rev. A 59(6), 4836-4839 (1999).
[CrossRef]

M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, "Faraday spectroscopy in a dark optical trap," Phys. Rev. A 77, 063417 (2008).
[CrossRef]

M. L. Terraciano, S. E. Olson, M. Bashkansky, Z. Dutton, and F. K. Fatemi, "Magnetically controlled velocity selection in a cold-atom sample using stimulated Raman transitions," Phys. Rev. A 76, 053421 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, "Atomic velocity selection using stimulated Raman transitions," Phys. Rev. Lett. 66(18), 2297-2300 (1991).
[CrossRef] [PubMed]

M. Vengalattore, J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Sadler, and D. M. Stamper-Kurn, "High-Resolution Magnetometry with a Spinor Bose-Einstein Condensate," Phys. Rev. Lett. 98, 200801 (2007).
[CrossRef] [PubMed]

Technical Physics (1)

E. B. Alexandrov, M. V. Balabas, A. K. Vershovski, and A. S. Pazgalev, "Experimental demonstration of the sensitivity of an optically pumped quantum magnetometer," Technical Physics 49, 779-783 (2004).
[CrossRef]

Other (1)

D. D. McGregor, "High-sensitivity helium resonance magnetometers," Review of Scientific Instruments 58(6), 1067-1076 (1987). URL http://link.aip.org/link/?RSI/58/1067/1.
[CrossRef]

Supplementary Material (2)

» Media 1: AVI (278 KB)     
» Media 2: AVI (451 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

(a) Lab-frame energy level diagram for velocity-selective resonances in 85Rb. (b) Schematic picture of imaging principle in a uniform field. Resonant velocity classes, at v=vavg ±νr (with vavg as defined in Eq. 4 for Δm=0,±1 transitions), are drawn in red for atoms emanating from a point source. The photon momentum kicks give these atoms average velocity vavg , leading to increased fluorescence at specific locations.

Fig. 2.
Fig. 2.

Experimental setup for magnetic field imaging. A retroreflected Raman beam propagates along the x-axis to form a lin⊥lin field. A CCD along the y-axis images fluorescence. Gravity is out of the plane of the figure. Helmholtz bias coils are used along each axis.

Fig. 3.
Fig. 3.

a) Images of expanded atom cloud in the presence of a uniform magnetic field for different bias field values. dBz /dI=1.5G/A and Bz =0 at I=243mA.

Fig. 4.
Fig. 4.

Top: Background-subtracted images along the y-axis for different magnetic field gradients. (a) Experiment (Media 1). (b) Simulations. Bottom: (c) Signal integrated along radial cuts through the experimental images as a function of angle from the vertical.

Fig. 5.
Fig. 5.

(a) Experimental images of the atom cloud for different values of the x-directed bias field, in the presence of a slight gradient (B′=0.5G/cm) (Media 2). Bx =0 at I=500mA, and dBx /dI=-0.844G/A. (b) Simulations using the same conditions. The difference in visibility between simulation and experiment is explained in the text.

Fig. 6.
Fig. 6.

Plot of the squares of the two sides of Eq. 9 showing the bands over which resonance occurs. For bias fields along k (), the stripe features have asymmetric widths. The shaded parabola (square of l.h.s. of Eq. 9) is thickened because the resonance occurs over a small band of velocities.

Equations (11)

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

E i = p i 2 2 M + m i h ¯ ω L + E i LS
E f = ( p i ± 2 h ¯ k ) 2 2 M + m f h ¯ ω L + E f LS
( p avg h ¯ k ) 2 2 M + m i h ¯ ω L + E i LS = ( p avg ± h ¯ k ) 2 2 M + m f h ¯ ω L + E f LS
2 k · v avg = ± Δ m ω L
B AH ( r ) = 0.5 B ( x x ̂ + y y ̂ 2 z z ̂ )
2 k v x = 2 k x T r = ± ( 0.5 ω L x ) 2 + ( 0.5 ω L y ) 2 + ( ω L z ) 2
m xz = m xy 2 = ± 1 α 2 2 α
θ xz = 2 θ xy = ± 2 α = ± ω L T r 2 k
2 k v x = 2 k x T r = ( ω L 0 x + 0.5 ω L x ) 2 + ( 0.5 ω L y ) 2 + ( ω L z ) 2
x c = 2 ω L 0 x ω L α 2 1 α 2
D = 2 ω L 0 x ω L α 1 α 2 = x c α

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