Linear and angular momenta in tightly focused vortex
segmented beams of light
(Invited Paper)
Martin Neugebauer
1,2
, Andrea Aiello
1,2
, and Peter Banzer
1,2,
*
1
Max Planck Institute for the Science of Light, Staudtstr. 2, Erlangen D-91058, Germany
2
Institute of Optics, Information and Photonics, University Erlangen-Nuremberg, Staudtstr. 7/B2,
Erlangen D-91058, Germany
*Corresponding author: peter.banzer@mpl.mpg.de
Received October 31, 2016; accepted January 6, 2017; posted online February 6, 2017
We investigate the linear momentum density of light, which can be decomposed into spin and orbital parts, in the
complex three-dimensional field distributions of tightly focused vortex segmented beams. The chosen angular
spectrum exhibits two spatially separated vortices of opposite charge and orthogonal circular polarization to
generate phase vortices in a meridional plane of observation. In the vicinity of those vortices, regions of negative
orbital linear momentum occur. Besides these phase vortices, the occurrence of transverse orbital angular
momentum manifests in a vortex charge-dependent relative shift of the energy density and linear momentum
density.
OCIS codes: 260.5430, 260.1960.
doi: 10.3788/COL201715.030003.
The linear and angular momenta (LM and AM) of light
are of great importance in the description and understand-
ing of dynamical processes concerning light-matter inter-
action phenom ena. Representative examples are optical
manipulation experiments, which demonstrate the pos-
sibility of transferring LM and AM from incoming light
to trapped particles
[1–3]
. For instance, spin AM results in
a torque, while LM corresponds to a force or radiation
pressure acting on a Rayleigh scatterer
[4,5]
. A deep under-
standing of this type of light-matter interaction is particu-
larly important in highly confined and/or structured
electromagnetic fields
[6–8]
. Recently, this led to a growing
interest in the theoretical description
[5,9–11]
and measure-
ment
[12,13]
of the individual constituents of LM and AM
occurring in various nano-optical systems. In general, it
is convenient to define these dynamical quantities locally,
when complex three-dimensional field distributions are
considered. Hence, we introduce the cycl e-averaged LM
density, p ¼ Re½E
× H∕c
2
, which is proportional to the
energy flow described by the Poynting vector, where
the electric and magnetic fields E and H, respectively,
are supposed to be monochromatic with an angular
frequency ω. The LM density can be separated into two
distinct parts representing the spin (s) and orbital (o)
contributions
[5–11]
,
p ¼ p
s
þ p
o
: (1)
The spin LM density p
s
, sometimes referred to as
Belinfante’s spin momentum density
[5,12–15]
, is defined by
the curl of the spin AM density s,
s ¼ Im½ϵ
0
E
× E þ μ
0
H
× H∕4ω; (2)
p
s
¼ ∇ × s∕2; (3)
where ϵ
0
and μ
0
are the permittivity and permeability of
the vacuum, respectively. In contrast, the orbital LM
density is related to the phase gradients of all individual
field components, weighted by their corresponding energy
densities:
p
o
¼ Im½ϵ
0
E
·ð∇ÞE þ μ
0
H
·ð∇ÞH∕4ω: (4)
Here, the notation A·ð∇ÞB ¼ A
x
∇B
x
þ A
y
∇B
y
þ A
z
∇B
z
is used. From Eqs. (2)–(4) it follows that both parts
of the LM density can again be separated into electric
and magnetic components
[9]
, with p
o
¼ p
oE
þ p
oH
and
p
s
¼ p
sE
þ p
sH
.
In a simple single plane wave scenario, p is by definition
parallel to the propagation direction, while s can be both,
parallel, or antiparallel. In this case, since s is independent
of the position; it follows that p
s
¼ 0, and p
o
¼ p.
However, in highly confined field distributions more
complex phenomena can occur. For example, regions of
negative energy flow or regions where p is antiparallel with
respect to the global propagation direction of the light
field occur in the waist of tightly focused beams
[16,17]
,in
edge-diffracted fields
[18]
, and in the case of total internal
reflection
[19–21]
. Typically, these negative energy flows are
linked to the occurrence of transverse vortices
[17–22]
. Addi-
tionally, a negative longitudinal component of p
s
, which is
linked to the occurrence of transverse components of s, has
been demonstrated in various optical systems consisting of
propagating
[12,14]
or evanescent waves
[5]
.
In this work, we investigate the occurrence of a negative
longitudinal component of p
o
, a phenomenon sometimes
referred to as ‘backflow’
[21,23–25]
, in tightly focused
composite beams consisting of two spatially separated vor-
tices with opposite charge and circular polarization of op-
posite handedness. Similar to a spin segmented beam
[26,27]
,
COL 15(3), 030003(2017) CHINESE OPTICS LETTERS March 10, 2017
1671-7694/2017/030003(5) 030003-1 © 2017 Chinese Optics Letters
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