Biomedical Engineering Reference
In-Depth Information
Figure 10.2
Intensity profiles of the Laguerre-Gaussian modes.
The amplitude
u
pl
(
r
ϕ
z
) of the Laguerre-Gaussian mode can be
written as
√
2
r
ω
(
z
)
|
l
|
L
|
l
p
2
r
2
exp(
−
il
ϕ
)
1
ω
(
z
)
u
p
,
l
(
r
,
ϕ
,
z
)
=
C
p
,
l
ω
2
(
z
)
×
exp
exp
exp[
i
(2
p
+ |
l
| +
1)
φ
(
z
)],
r
2
ikr
2
2
R
(
z
)
−
−
ω
2
(
z
)
(10.6)
where
L
|
l
p
are generalized Laguerre polynomials,
p
is the radial
index,
l
is the azimuthal index (known as the topological charge)
ω
φ
(
z
)isthe
phase parameter, and
C
p
,
l
is the normalization constant. The spiral
wavefront of OV is visualized by fringes formed by interference
between the OV and a spherical (or plane) reference wavefront. The
sphericalreferencedfringesasshowninFig.10.3hadasinglespiral
indicating the existenceof a phase singularity [25, 26].
(
z
)isthebeamspotsize,
R
(
z
)istheradiusofcurvature,
Figure 10.3
Spatial forms of OV with
|
l
|
=
1. (a) Intensity profile of the
OV. Spherical referenced fringes with (b) clockwise and (c) anti-clockwise
directions. Reprinted from Ref. [27] with permission of the Optical Society,
Copyright 2012.
