Biomedical Engineering Reference
In-Depth Information
where the variables (
ν
r
,
ν
θ
) are Fourier conjugate of the direct-space
cylindricalcoordinates(
r
,
θ
)usedtorepresenttheVogelspiralarray,
α
is the irrational divergence angle, a
0
is a constant scaling factor,
and
N
is the number of particles in the array [51].
Fourier-Hankel modal decomposition can be used to analyze a
superposition state of OAM states carrying modes in the far-field
pattern and determine their relative contribution to the overall
diffracted beam. Decomposition of
) into a basis [52, 64] set
with helical phase fronts is accomplished through Fourier-Hankel
decomposition (FHD) according to:
ρ
(
r
,
θ
∞
2
π
1
2
)
J
m
(
k
r
r
)
e
im
θ
=
θρ
θ
f
(
m
,
k
r
)
rdrd
(
r
,
(11.5)
π
0
0
where
J
m
is the
m
-th order Bessel function. In this decomposition,
the
m
-th order function identifies OAM states with azimuthal
number
m
, accommodating both positive and negative integer
m
values.
By analytically performing FHD analysis, Dal Negro et al. [51]
demonstrated that diffracted optical beams by Vogel spirals carry
OAM values arranged in aperiodic numerical sequences determined
by the number-theoretic properties of the irrational divergence
angle
α
. More precisely, the OAM values transmitted in the far-field
region are directly determined by the rational approximations of
the continued fraction expansion of the irrational divergence angles
of Vogel spirals [51]. In particular, wave diffraction by GA arrays
generates a Fibonacci sequence of OAM values in the Fraunhofer
far-field region. This fascinating property of Vogel spirals can be
understoodclearlybyconsideringtheanalyticalsolutionoftheFHD
of the far-field radiation pattern, givenby [51]:
N
A
(
k
r
)
e
imn
α
f
(
m
,
k
r
)
=
(11.6)
n
=
1
where
A
(
k
r
)isa
k
r
-dependent coe
cient, which can be ignored
as we are concerned with the azimuthal dependence contained in
f
(
m
).
We see from the result in Eq. (11.6) that when the product
m
α
is an integer, the
N
contributing waves will be exactly in phase
