We address the question of whether there exists a hidden relationship between the near-field distribution generated by an oscillating electric dipole and the so-called cross-polarization of a collimated beam of light. We find that the answer is affirmative by showing that the complex field distributions occurring in both cases have a common physical origin: the requirement that the electromagnetic fields must be transverse.

1.
H.
Kubota
and
S.
Inoué
, “
Diffraction images in the polarizing microscope
,”
J. Opt. Soc. Am.
49
,
191
198
(
1959
).
2.
A. C.
Ludwig
, “
The definition of cross polarization
,”
IEEE Trans. Antennas Propag.
21
,
116
119
(
1973
).
3.
M.
Lax
,
W. H.
Louisell
, and
W. B.
McKnight
, “
From Maxwell to paraxial wave optics
,”
Phys. Rev. A
11
,
1365
1370
(
1975
).
4.
L. W.
Davis
, “
Theory of electromagnetic beams
,”
Phys. Rev. A
19
,
1177
1179
(
1979
).
5.
D. N.
Pattanayak
and
G. P.
Agrawal
, “
Representation of vector electromagnetic beams
,”
Phys. Rev. A
22
,
1159
1164
(
1980
).
6.
R.
Simon
,
E. C. G.
Sudarshan
, and
N.
Mukunda
, “
Cross polarization in Laser beams
,”
Appl. Opt.
26
,
1589
1593
(
1987
).
7.
W. L.
Erikson
and
S.
Singh
, “
Polarization properties of Maxwell-Gaussian laser beams
,”
Phys. Rev. E
49
,
5778
5786
(
1994
).
8.
V. V.
Tuchin
,
L. V.
Wang
, and
D. A.
Zimnyakov
,
Optical Polarization in Biomedical Applications
(
Springer
,
Heidelberg
,
2006
).
9.
K.
Claborn
,
E.
Puklin-Faucher
,
M.
Kurimoto
,
W.
Kominsky
, and
B.
Kahr
, “
Circular dichroism imaging microscopy: application to Enantiomorphous Twinnings in 1,8-Dihydroxyanthraquinone
,”
J. Am. Chem. Soc.
125
,
14825
14831
(
2003
).
10.
R. A.
Carlton
,
Pharmaceutical Microscopy
(
Springer
,
Heidelberg
,
2011
), Chap. 2.
11.
For a brief description of how polarization can be used for 3D movie technology, see
Matt
Cowan
, “REAL D 3D Theatrical System: A Technical Overview,” <http://www.edcf.net/edcf_docs/real-d.pdf> (
2007
). A description of the U.S. patent for this method can be found at <http://worldwide.espacenet.com/publicationDetails/biblio?CC=US&NR=6975345&KC=&FT=E&locale=en_EP>.
12.
J. A.
Stratton
,
Electromagnetic Theory
, IEEE Press Series on Electromagnetic Wave Theory (
John Wiley & Sons, Inc.
,
Hoboken, NJ
,
2007
), Chap. 1.
13.
J. D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
Hoboken, NJ
,
1999
).
14.
E. A.
Essex
, “
Hertz vector potentials of electromagnetic theory
,”
Am. J. Phys.
45
,
1099
1101
(
1977
).
15.
In this work, we consider the fields emitted by an oscillating dipole only. In principle, one could consider a more complicated oscillator characterized by quadrupoles and higher-order terms. However, as stressed in Sec. 9.3 of Ref. 13, this would require more elaborate mathematics. We believe that the use of the latter may obscure the physical significance of the problem studied here. Therefore, in the present work, we merely note that considering multipole radiation would enable a comparison with beams of light with non-uniform polarization patterns, like radially and azimuthally polarized beams; see, e.g.,
A.
Holleczek
,
A.
Aiello
,
C.
Gabriel
,
Ch.
Marquardt
, and
G.
Leuchs
, “
Classical and quantum properties of cylindrically polarized states of light
,”
Opt. Exp.
19
,
9714
9736
(
2011
). A thorough didactic treatment of this issue would require an independent, additional paper.
16.
W. J. M.
Kort-Kamp
and
C.
Farina
, “
On the exact electric and magnetic fields of an electric dipole
,”
Am. J. Phys.
79
,
111
114
(
2011
).
17.
Reference 13, pp. 407–413.
18.
Y.
Fainman
and
J.
Shamir
, “
Polarization of nonplanar wave fronts
,”
Appl. Opt.
23
,
3188
3195
(
1984
).
19.
E.
Hecht
,
Optics
, 4th ed. (
Addison-Wesley
,
San Francisco
,
2002
), Chap. 8.
20.
L.
Mandel
and
E.
Wolf
,
Optical Coherence and Quantum Optics
(
Cambridge U.P.
,
New York
,
1995
), Chap. 10.
21.
It is instructive for the reader to note the similarity between Eq. (22) and the expression for the quantum mechanical expectation value of the momentum operator in terms of the wave function in the momentum representation. See, e.g., Eq. (3.26) in
E.
Merzbacher
,
Quantum Mechanics
, 3rd ed. (
Wiley
,
Hoboken, NJ
,
1988
).
22.
M. A.
Porras
,
J.
Alda
, and
E.
Bernabeu
, “
Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams
,”
Appl. Opt.
31
,
6389
6402
(
1992
).
23.
A. E.
Siegman
,
Lasers
(
University Science Books
,
Sausalito, CA
,
1986
), Chap. 17.
24.
F.
Rohrlich
, “
The validity of the Helmholtz theorem
,”
Am. J. Phys.
72
,
412
413
(
2004
).
25.
C.
Cohen-Tannoudji
,
J.
Dupont-Roc
, and
G.
Grynberg
,
Photons & Atoms
(
Wiley-VCH
,
Weinheim
,
2004
), Chap. 1.
26.
J. M.
Aguirregabiria
,
A.
Hernandez
, and
M.
Rivas
, “
δ-function converging sequences
,”
Am. J. Phys.
70
,
180
185
(
2002
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.