Moment of momentum is not spin. 2
- From: khrapko_ri@xxxxxxxxxxx
- Date: Sun, 7 Dec 2008 02:59:35 -0800 (PST)
---This is a submittion to CLEO/IQEC-2009 conference cleo@xxxxxxx (see
also www.sciprint.org, Spin_dipole_radiation)
---A circularly polarized light beam carries an angular momentum (AM)
[1,2]. However, troubling questions exist: what is the distribution of
this AM over the beam section, and what is the nature of the AM,
orbital or spin?
---A paraxial circularly polarized Laguerre-Gaussian beam [3], is an
eigenfunction of the orbital, not spin, AM operator. This means that
both, the circular polarization and the spiral phase front related
with l, carry only orbital AM, not spin, in the frame of the standard
electrodynamics.
---Now we consider an exact, not paraxial, solution of the Maxwell
equations; the solution for the radiation of a rotating electric
dipole [4-6] in the spherical coordinates.
---An angular distribution of the energy flux is proportional to
cos^2\theta+1, and an angular distribution of z-component of the AM
flux, i.e., of torque, is proportional to sin^2\theta. We present also
a distribution of the degree of circular polarization of the radiation
[4], which approximately equals the ratio of lengths of the axes of
the ellipse, i.e. cos\theta.
---It is seen that the AM is emitted mainly into the equatorial part
of space, situated near the x-y-plane where the polarization is
elliptic or linear. Polar regions, situated near the z-axis, are
scanty by the AM, although they are intensively illuminated by the
almost circularly polarized radiation. So, if we associate spin of an
electromagnetic radiation with a circular polarization, we must
recognize the AM is an orbital AM, not spin. Also note, fields of the
radiation of a rotating electric dipole [4-6] are eigenfunctions of
the orbital, not spin, AM operator. This confirms the orbital nature
of the AM.
---Thus we must recognise the standard electrodynamics cannot catch
sight of spin of electromagnetic fields, and it is in need of an
expansion.
---The classical field theory points the way to the expansion. The
Lagrange formalism gives two divergence-free tensors for free fields,
energy-momentum and spin tensors [7]:
---Unfortunately, the standard Belinfante-Rosenfeld procedure [8,9]
eliminates the spin tensor of electrodynamics [10,11]. So, we proposed
an alternative procedure [12,13], which gives the Maxwell energy-
momentum tensor and an elecrtrodynamics' spin tensor. This spin tensor
yields an angular distribution of z-component of the spin flux in the
rotating electric dipole radiation [5,6], which is proportional to
cos^2\theta and the total flux of z-component of the spin, which is
half of the total orbital angular momentum flux. However, the ratio of
the spin flux density to the power density at \theta=0 equals 1/omega
just as for a photon because the radiation is circularly polarized
with plane phase front along z-axis:
------The idea of an electrodynamics' spin tensor was rejected approx
400 times by 30 scientific journal since May 12, 1998.
[1] R. A. Beth, Direct Detection of the angular momentum of light,
Phys. Rev. 48, 471 (1935).
[2] S. Parkin, G. Knoner, T. A. Nieminen, N. R. Heckenberg and H.
Rubinsztein-Dunlop, Measurement of the total optical angular momentum
transfer in optical tweezers, Optics Express 14, 6963 (2006).
[3] L. Allen, M. J. Padgett, M. Babiker, The orbital angular momentum
of light, in Progress in Optics XXXIX (Elsevier, Amsterdam, 1999) p.
298
[4] A. Corney, Atomic and Laser Spectroscopy (Oxford University Press,
1977).
[5] R. I. Khrapko. Radiation of spin by a rotator,
mp_arc@xxxxxxxxxxxxxxxxxx 03-315 (2003)
[6] R. I. Khrapko, A rotating electric dipole radiates spin and
orbital angular momentum, www.sciprint.org (2006)
[7] F. Rohrlich, Classical Charged Particles (Addison-Wesley, Mass.
1965)
[8] F. J. Belinfante, Physica 6, 887 (1939).
[9] L. Rosenfeld, Memoires de l'Academie Royale des Sciences de
Belgiques 8 No 6 (1940).
[10] R. I.Khrapko, Mechanical stresses produced by a light beam, J.
Modern Optics 55, 1487-1500 (2008)
[11] R. I.Khrapko, Mechanical stresses produced by a light beam,
http://www.sciprint.org (2007)
[12] R. I. Khrapko, True energy-momentum tensors are unique.
Electrodynamics spin tensor is not zero, physics/0102084
[13] R. I. Khrapko. Violation of the gauge equivalence, physics/
0105031
Radi Khrapko
.
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