Biomedical Engineering Reference
In-Depth Information
fact, for all integer Bessel orders
m
=
q
n
, the exponential sum in
Eq. (11.6) will give in-phase contributions to the FHD and produce
strongazimuthalpeaks.Therefore,oncetherationalapproximations
of the irrational number
ζ
have been identified on the basis of
the well-established continued fraction theory, Vogel spirals that
“encode” in their OAM spectra well-defined numeric sequence of
azimuthal orders, specified by the denominators
q
n
in Eq. (11.8),
can be readily designed. This general strategy enables to generalize
the OAM Fibonacci coding recently discovered for GA spirals. It
is important to realize that using the approach just introduced,
it becomes possible to generate structured light with controllable
OAM spectra by plasmon-enhanced diffraction in aperiodic arrays
of nanoparticles. The use of very compact Vogel spiral arrays
additionally enables the generation of large OAM azimuthal orders
that are intrinsicto their geometric structures.
InFig.11.9,weshowthecalculatedFraunhoferfar-fieldsandthe
correspondingazimuthalOAMspectraofGAand
π
spirals.Asweare
primarily concerned with the azimuthal component
f
(
m
)ofOAM,
we can sum
f
(
m
,
k
r
) over radial the wavenumbers
k
r
.
Figure11.9c,ddemonstratetheveryrichstructureofOAMpeaks
ofthescatteredradiationenabledbytheaperiodicgeometrycontrol
of Vogel spirals. These peaks exactly occur at azimuthal numbers
(labeledinthefigures)correspondingtothedifferentdenominators
of the rational approximations of the irrational divergence angles
used to generate the spirals, as previously discussed. In particular,
theFibonaccisequenceofOAMvaluesiscodedinthefar-fieldregion
of the radiation scattered by the GA spiral (Fig. 11.9c).
Using phase-delayed interferometric measurements, Lawrence
et al. [47] have recently measured experimentally the complex
electric field distribution of scattered radiation by Vogel spirals
and demonstrated for the first time structured light carrying
multiple values of OAM in the far-field region of arrays of metallic
nanoparticles, in excellent agreement with the analytical theory
[51]. In addition, direct excitation of optical modes carrying OAM
values has recently been demonstrated in light-emitting dielectric
GA spiral structures consisting ofEr-doped SiN nanopillars [46].
Beforediscussingmoreindepththeexperimentaldemonstration
of coded OAM spectra with large azimuthal values, we will briefly
