Biomedical Engineering Reference
In-Depth Information
increasing along this path. This difference corresponds to another
2
π
phase retardation in view of the SPPs in the same rotational
direction of the incident field. Consequently, the phase distribution
alongtheazimuthwillpossessatotalamountof4
π
phasedelay.This
two-timesofthecycleinthephasedistributionisrepresentedinthe
form of agray-leveled circle near the originpoint of Fig. 7.2(a).
On the contrary, when an LCP light is incident on this RH spiral,
as in Fig. 7.2(b), the magniude of the geometrical phase delay of
2
π
, which is added to the SAM-oriented phase delay, is in the
opposite direction to the direction of SAM. Therefore, the azimuthal
dependence of the SPPs near the origin cancels out and the phase
distribution near the originis constant as shown in Fig. 7.2(b).
To appropriately describe such a SAM-OAM conversion, it is
convenient to adopt the convention of a topological charge number
[26, 27]. The topological charge number is defined as the number
corresponding to the phase modulation along a circular path near
the singular point of the field divided by 2
π
. When a light field is
scattered by matter with a geometrical helicity, it is known that the
topological charge number
m
T
can be expressed as
m
T
=
m
O
+
m
SO
+
m
G
, (7.1)
where
m
O
,
m
SO
,and
m
G
are the AM numbers coming from
OAM carried by the incident light, SAM-OAM conversion, and the
geometrical helicity, respectively. As these numbers can be positive,
zero, or negative,
m
T
can be larger or smaller than
m
O
according to
therelationbetweenthegeometryofthescattererandthespinstate
of the incident field.
Using the convention of topological charge number, it is easy to
describethecasesinFigs.7.1and7.2.Fortheseexamples,theresults
given by Eq. (7.1) are as follows:
m
T
=
m
O
+
m
SO
+
m
G
=
0
−
1
+
0
=−
1 (Fig. 7.1), (7.2)
m
T
=
m
O
+
m
SO
+
m
G
=
0
+
1
+
1
=
2 [Fig. 7.2(a)], (7.3)
m
T
=
m
O
+
m
SO
+
m
G
=
0
−
1
+
1
=
0 [Fig. 7.2(b)]. (7.4)
The AM numbers in Eq. (7.1) are not necessarily integers.
They can be rational or irrational numbers. Even worse, for some
complicated geometries, the AM number
m
G
cannot be identified in
a simple manner. In the following sections, we will examine such
examples and investigate the effect arising in cases of geometrically
complicated patterns.
