Abstract

We derive the magnetic dipole selection rules and the magnetic dipole absorption rate for a spherical semiconductor quantum dot. We find that electric dipole and magnetic dipole transitions are exclusive and therefore can be spectrally distinguished. The magnitudes of electric and magnetic absorption rates are compared for excitation with a strongly focused azimuthally polarized beam. It turns out that spatial optical resolution can be increased by detection of the ratio of magnetic and electric absorption rates. Resolution is limited only by the purity of the laser mode used for excitation.

© 2002 Optical Society of America

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References

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  1. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Dover, Mineola, New York, 1998).
  2. J. R. Zurita-Sánchez and L. Novotny, “Multipolar interband absorption in a semiconductor. I. Electric quadrupole enhancement,” J. Opt. Soc. Am. B 19, 1355–1362 (2002).
    [CrossRef]
  3. B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, “Optical near-field response of semiconductor quantum dots,” Phys. Rev. B 55, 13715–13725 (1997).
    [CrossRef]
  4. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1993).
  5. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef] [PubMed]
  6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef] [PubMed]

2002 (1)

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

2000 (1)

1997 (1)

B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, “Optical near-field response of semiconductor quantum dots,” Phys. Rev. B 55, 13715–13725 (1997).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

Hanewinkel, B.

B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, “Optical near-field response of semiconductor quantum dots,” Phys. Rev. B 55, 13715–13725 (1997).
[CrossRef]

Knorr, A.

B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, “Optical near-field response of semiconductor quantum dots,” Phys. Rev. B 55, 13715–13725 (1997).
[CrossRef]

Koch, S. W.

B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, “Optical near-field response of semiconductor quantum dots,” Phys. Rev. B 55, 13715–13725 (1997).
[CrossRef]

Novotny, L.

J. R. Zurita-Sánchez and L. Novotny, “Multipolar interband absorption in a semiconductor. I. Electric quadrupole enhancement,” J. Opt. Soc. Am. B 19, 1355–1362 (2002).
[CrossRef]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Thomas, P.

B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, “Optical near-field response of semiconductor quantum dots,” Phys. Rev. B 55, 13715–13725 (1997).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

Zurita-Sánchez, J. R.

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

Opt. Express (1)

Phys. Rev. B (1)

B. Hanewinkel, A. Knorr, P. Thomas, and S. W. Koch, “Optical near-field response of semiconductor quantum dots,” Phys. Rev. B 55, 13715–13725 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Other (2)

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1993).

D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Dover, Mineola, New York, 1998).

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Figures (4)

Fig. 1
Fig. 1

Excitation of a spherical quantum dot (QD) with a focused azimuthally polarized beam. An objective (OB) with focal distance f [distance between rear principal plane (PP) and focal plane (FP)] and numerical aperture N.A.=n sin θmax focuses an incoming azimuthally polarized beam through an exit pupil (EP) along the optical axis (OA) on the focal plane. The quantum dot is located in the focal plane at r0=ρ0nρ.

Fig. 2
Fig. 2

Electric field |E˜|2 (solid curves) and magnetic field |B˜|2 (dashed curves) of a focused azimuthally polarized beam evaluated in the focal plane as a function of the radial coordinate ρ0 for numerical apertures N.A.=1.0,1.3,1.4,2.9. The electric field is zero at the center, whereas the magnetic field strength increases as the N.A. becomes larger. The wavelength is λ=532 nm; the filling factor is f0=1.

Fig. 3
Fig. 3

Ratio of the magnetic dipole and electric dipole absorption rates (αE/αM) as a function of the radial coordinate ρ0 of the quantum dot. The left (right) vertical axis corresponds to quantum dot radius a=5 nm (a=10 nm). The ratio corresponding to plane-wave excitation is indicated by the horizontal line.λ=532 nm,fo=1.

Fig. 4
Fig. 4

Inverse ratio (αM/αE) as a function of the radial coordinate ρ0 of a quantum dot with a=5 nm and a=10 nm. In a length interval of a few nanometers, the magnetic dipole absorption rate is stronger than the electric dipole absorption rate. The ratio is largely independent of N.A., but a large N.A. is required for a good signal-to-noise ratio. λ=532 nm, f0=1.

Equations (26)

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B(r, t)=B˜(r)exp(-iωt)+c.c.
HˆM=-B(0, t)·e2mo rˆ×pˆ,
HˆM=ie2mo B(0, t)·Ψˆ(r)(r×Ψˆ)d3r.
HˆM=ie2mo B(0, t)·nlmrstfˆnlmgˆrst×1Vo uc,0*(r)ζnlme*(r)r×[uv,0(r)ζrsth(r)]d3r+h.c.,
HˆM=ie2mo B(0, t)·nlmrstqfˆnlmgˆrst×1Vo Vouc,0*(r+Rq)ζnlme*(r+Rq)×(r+Rq)[uv,0(r+Rq)ζrsth(r+Rq)]×d3r+h.c.,
HˆMie2mo B(0, t)·nlmrstqfˆnlmgˆrstMcvζnlme*(Rq)ζrsth(Rq)+h.c.,
Mcv=1Vo Vouc,0*(r)(Rq+r)×uv,0(r)d3r.
Mcv=Rq×mcv,
mcv1Vo Vouc,0*(r)uv,0(r)d3r.
HˆM=ie2mo B(0, t)·nlmrstfˆnlmgˆrstDnmlrst×mcv+h.c.,
Dnmlrstζnlme*(R)Rζrsth(R)d3R.
αM=2π nlmrst|nml; rst|HˆintM|0|2×δ[ω-(Enle+Ersh)],
αM=2π 2e24mo2 nlmrst|B˜(0)·(Dnmlrst×mcv)|2×δ[ω-(Enle+Ersh)].
m-t=±1,l-s=±1,
m-t=0,l-s=±1.
E˜(ρ, φ)=AI1(ρ)nφ,
I1(ρ)0θmaxfw(θ)(cos θ)3/2(sin2 θ)J1(kρ sin θ)dθ,
fw(θ)exp-1fo2 sin2 θsin2 θmax,
fowof sinθmax.
B˜(ρ)=-Anc[I2(ρ)nρ-4iI3(ρ)nz],
I2(ρ)0θmaxfw(θ)(cos θ)1/2(sin2 θ)J1(kρ sin θ)dθ,
I3(ρ)0θmaxfw(θ)(cos θ)1/2(sin3 θ)Jo(kρ sin θ)dθ.
αM=2π 2e22mo2|B˜(0)|2|D|2|m|2δ[ω-(E10e+E11h)],
|D|=|D100110|=|D100111|=|D10011-1|,
|m|=|mcv1|=|mcv2|=|mcv3|.
αMαE=(2πc)22λ2|D|2 |B˜|2|E˜|2.

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