Abstract

In this work, scattering of an incident electric field from a moving atom is reexamined classically in two steps: the time-dependent current density created by the field inside the atom is first calculated under the electric-dipole approximation, and is then used to calculate the field scattered from the atom. Unlike the conventional frame-hopping method, the present method does not need to treat the Doppler effect as an effect separated from the scattering process, and it derives instead of simply uses the Doppler effect.

© 2012 Optical Society of America

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References

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  1. J. C. Leader, “An analysis of the frequency spectrum of laser light scattered from moving rough objects,” J. Opt. Soc. Am. 67, 1091–1098 (1977).
    [CrossRef]
  2. B. Cairns and E. Wolf, “Changes in the spectrum of light scattered by a moving diffuser plate,” J. Opt. Soc. Am. A 8, 1922–1928 (1991).
    [CrossRef]
  3. P. De Cupis, “Relativistic scattering from moving fractally corrugated surfaces,” Opt. Lett. 28, 849–850 (2003).
    [CrossRef]
  4. X. Zhang and J. Yang, “Moving object detection based on shape prediction,” J. Opt. Soc. Am. A 26, 342–349 (2009).
    [CrossRef]
  5. W. Guo, “Multiple scattering of a plane scalar wave from a uniform dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
    [CrossRef]
  6. W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
    [CrossRef]
  7. W. Guo, “Temperature dependence of superluminal and subluminal propagation,” J. Opt. Pure Appl. Opt. 9, 1030–1033 (2007).
    [CrossRef]
  8. W. Guo and Y. Aktas, “Reexamination of the Doppler effect through Maxwell’s equations,” J. Opt. Soc. Am. A 29, 1568–1570 (2012).
    [CrossRef]
  9. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).
  10. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989).
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  12. W. Guo, “Optical response of two-electron atoms: a classical formulation,” Am. J. Phys. 75, 821–823 (2007).
    [CrossRef]
  13. O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
    [CrossRef]
  14. In principle, the electron should additionally experience a damping force to account for the fact that as the scattered field is emitted, the electron must lose its energy; see [9], for example. But, since this force is not essential in the present discussion, it is ignored for simplicity.
  15. B. Rossi, Optics (Addison-Wesley, 1957).
  16. Alternatively, the scattered field E⃗s can also be calculated by applying the Liénard–Wiechert relations [9] More specifically, the relations are first used to find the electric fields due to the electron and rest part of the atom. The electric fields are then added and simplified, with the help of the electric-dipole approximation, to get E⃗s.
  17. W. Guo, “Effects of back-action on the measurement of a sub-wavelength separation in the near-field region,” J. Opt. 13, 075704 (2011).
    [CrossRef]
  18. O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
    [CrossRef]

2012 (1)

2011 (1)

W. Guo, “Effects of back-action on the measurement of a sub-wavelength separation in the near-field region,” J. Opt. 13, 075704 (2011).
[CrossRef]

2009 (1)

2007 (3)

W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
[CrossRef]

W. Guo, “Temperature dependence of superluminal and subluminal propagation,” J. Opt. Pure Appl. Opt. 9, 1030–1033 (2007).
[CrossRef]

W. Guo, “Optical response of two-electron atoms: a classical formulation,” Am. J. Phys. 75, 821–823 (2007).
[CrossRef]

2003 (1)

2002 (1)

W. Guo, “Multiple scattering of a plane scalar wave from a uniform dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
[CrossRef]

1999 (1)

1998 (1)

O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
[CrossRef]

1991 (1)

1977 (1)

Aktas, Y.

Cairns, B.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989).

De Cupis, P.

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989).

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989).

Guo, W.

W. Guo and Y. Aktas, “Reexamination of the Doppler effect through Maxwell’s equations,” J. Opt. Soc. Am. A 29, 1568–1570 (2012).
[CrossRef]

W. Guo, “Effects of back-action on the measurement of a sub-wavelength separation in the near-field region,” J. Opt. 13, 075704 (2011).
[CrossRef]

W. Guo, “Temperature dependence of superluminal and subluminal propagation,” J. Opt. Pure Appl. Opt. 9, 1030–1033 (2007).
[CrossRef]

W. Guo, “Optical response of two-electron atoms: a classical formulation,” Am. J. Phys. 75, 821–823 (2007).
[CrossRef]

W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
[CrossRef]

W. Guo, “Multiple scattering of a plane scalar wave from a uniform dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Keller, O.

O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
[CrossRef]

O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
[CrossRef]

Leader, J. C.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Rossi, B.

B. Rossi, Optics (Addison-Wesley, 1957).

Wolf, E.

Yang, J.

Zhang, X.

Am. J. Phys. (2)

W. Guo, “Optical response of two-electron atoms: a classical formulation,” Am. J. Phys. 75, 821–823 (2007).
[CrossRef]

W. Guo, “Multiple scattering of a plane scalar wave from a uniform dielectric slab,” Am. J. Phys. 70, 1039–1043 (2002).
[CrossRef]

J. Opt. (1)

W. Guo, “Effects of back-action on the measurement of a sub-wavelength separation in the near-field region,” J. Opt. 13, 075704 (2011).
[CrossRef]

J. Opt. Pure Appl. Opt. (1)

W. Guo, “Temperature dependence of superluminal and subluminal propagation,” J. Opt. Pure Appl. Opt. 9, 1030–1033 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

W. Guo, “Corrections to Snell’s reflection law from atomic thermal motion,” Opt. Commun. 278, 253–256 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

O. Keller, “Propagator picture of the spatial confinement of quantum light emitted from an atom,” Phys. Rev. A 58, 3407–3425 (1998).
[CrossRef]

Other (6)

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, 1989).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

In principle, the electron should additionally experience a damping force to account for the fact that as the scattered field is emitted, the electron must lose its energy; see [9], for example. But, since this force is not essential in the present discussion, it is ignored for simplicity.

B. Rossi, Optics (Addison-Wesley, 1957).

Alternatively, the scattered field E⃗s can also be calculated by applying the Liénard–Wiechert relations [9] More specifically, the relations are first used to find the electric fields due to the electron and rest part of the atom. The electric fields are then added and simplified, with the help of the electric-dipole approximation, to get E⃗s.

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Equations (10)

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j⃗(r⃗,t)=j⃗0(t)δ(r⃗v⃗t),
md2dt2(v⃗t+R⃗)+mω02R⃗=eE⃗0eik⃗·(v⃗t+R⃗)iωt,
d2R⃗dt2+ω02R⃗=emE⃗0eiω(1k^·v⃗c1)temE⃗0eiω¯t,
R⃗=em(ω02ω¯2)E⃗0eiω¯t
j⃗(r⃗,t)=ie2ω¯m(ω¯2ω02)E⃗0eiω¯tδ(r⃗v⃗t).
2A⃗1c22A⃗t2=4πcj⃗.
A⃗(r⃗,t)=1cdr⃗1dt1j⃗(r⃗1,t1)|r⃗r⃗1|δ(t1+|r⃗r⃗1|ct)=ie2ω¯mc(ω¯2ω02)E⃗0dt1eiω¯t1|r⃗v⃗t1|δ(t1+|r⃗v⃗t1|ct),
A⃗(r⃗,t)ie2ω¯mc(ω¯2ω02)|r⃗|E⃗0dt1eiω¯t1(1r^·v⃗c1)1δ(t1+|r⃗|c1t1r^·v⃗c1)=ie2ω¯mc(ω¯2ω02)(1r^·v⃗c1)|r⃗|E⃗0eiω˜t+ik˜|r⃗|,
E⃗s(r⃗,t)=ik˜××A⃗(r⃗,t)=k˜2(r^×E⃗s)×r^eik˜|r⃗||r⃗|+[3r^(r^·E⃗s)E⃗s](1|r⃗|3ik˜|r⃗|2)eik˜|r⃗|k˜2(r^×E⃗s)×r^eik˜|r⃗||r⃗|,
ω¯=ω1k^·v⃗c11r^·v⃗c1,

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