9 Aperiodic Order in Nanoplasmonics
343
where the variables (τ r , τ θ ) are Fourier conjugate of the direct-space cylindrical coordinates (r, θ) used to represent the Vogel spiral density, γ is an irrational divergence
angle, a 0 is a constant scaling factor, and N is the number of particles in the array
[73].
A distinctive property of Vogel spiral nanoparticle arrays is that, when illuminated
by optical beams, they give rise to scattered radiation carrying OAM [69, 73]. Modal
decomposition can be used to analyze a superposition state of OAM carrying modes
in the far field pattern and determine their relative contribution to the overall field
[66, 72, 76]. Decomposition into a basis set with helical phase fronts, is accomplished
through Fourier–Hankel decomposition (FHD) according to:
f (m, k r ) =
1
2ϕ
√
0
2ϕ
0
rdrdΔΘ(r, Δ)J m (k r r )e
imΔ
(9.8)
where J m is the mth order Bessel function. In this decomposition, the mth order function carries OAM with azimuthal number m, accommodating positive and negative
integer values for m.
By analytically performing Fourier–Hankel Decomposition (FHD) analysis, Dal
Negro et al. [73] demonstrated that diffracted optical beams by Vogel spirals carry
OAM values arranged in aperiodic numerical sequences determined by the numbertheoretic properties of the irrational angle γ. In particular, wave diffraction by GA
arrays generates a Fibonacci sequence of OAM values in the Fraunhofer far field
region. More precisely, the OAM values transmitted in the far field region are directly
related to the rational approximations of the continued fraction expansion of the
irrational divergence angles of Vogel spirals [73].
It is important to realize that Vogel spiral arrays provide a very large spectrum of
OAM values relying uniquely on light scattering phenomena. In Fig. 9.6, we show
the calculated Fraunhofer far fields and the OAM azimuthal spectra of GA and
μ-spirals. Since we are primarily concerned with the azimuthal component f(m) of
OAM, we sum f (m, k r ) over radial the wavenumbers k r . Figures 9.6c, d demonstrate
the very rich structure of OAM peaks of the scattered radiation by Vogel spirals.
These peaks occur at azimuthal numbers (labeled in the figures) corresponding to
the denominators of the rational approximations of the irrational divergence angles
used to generate the spirals. The Fibonacci sequence of OAM values is coded in the
far field region of the radiation scattered by the GA spiral.
This fascinating property of Vogel spirals can be understood clearly by considering
the analytical solution of the FHD of the far field radiation pattern, which is given
by [73]:
f (m, k r ) =
N
n=1
A(k r )e
imnγ
(9.9)
where A(k r )is a k r -dependent coefficient, which can be ignored since we are concerned with the azimuthal dependence contained in f(m).
343
where the variables (τ r , τ θ ) are Fourier conjugate of the direct-space cylindrical coordinates (r, θ) used to represent the Vogel spiral density, γ is an irrational divergence
angle, a 0 is a constant scaling factor, and N is the number of particles in the array
[73].
A distinctive property of Vogel spiral nanoparticle arrays is that, when illuminated
by optical beams, they give rise to scattered radiation carrying OAM [69, 73]. Modal
decomposition can be used to analyze a superposition state of OAM carrying modes
in the far field pattern and determine their relative contribution to the overall field
[66, 72, 76]. Decomposition into a basis set with helical phase fronts, is accomplished
through Fourier–Hankel decomposition (FHD) according to:
f (m, k r ) =
1
2ϕ
√
0
2ϕ
0
rdrdΔΘ(r, Δ)J m (k r r )e
imΔ
(9.8)
where J m is the mth order Bessel function. In this decomposition, the mth order function carries OAM with azimuthal number m, accommodating positive and negative
integer values for m.
By analytically performing Fourier–Hankel Decomposition (FHD) analysis, Dal
Negro et al. [73] demonstrated that diffracted optical beams by Vogel spirals carry
OAM values arranged in aperiodic numerical sequences determined by the numbertheoretic properties of the irrational angle γ. In particular, wave diffraction by GA
arrays generates a Fibonacci sequence of OAM values in the Fraunhofer far field
region. More precisely, the OAM values transmitted in the far field region are directly
related to the rational approximations of the continued fraction expansion of the
irrational divergence angles of Vogel spirals [73].
It is important to realize that Vogel spiral arrays provide a very large spectrum of
OAM values relying uniquely on light scattering phenomena. In Fig. 9.6, we show
the calculated Fraunhofer far fields and the OAM azimuthal spectra of GA and
μ-spirals. Since we are primarily concerned with the azimuthal component f(m) of
OAM, we sum f (m, k r ) over radial the wavenumbers k r . Figures 9.6c, d demonstrate
the very rich structure of OAM peaks of the scattered radiation by Vogel spirals.
These peaks occur at azimuthal numbers (labeled in the figures) corresponding to
the denominators of the rational approximations of the irrational divergence angles
used to generate the spirals. The Fibonacci sequence of OAM values is coded in the
far field region of the radiation scattered by the GA spiral.
This fascinating property of Vogel spirals can be understood clearly by considering
the analytical solution of the FHD of the far field radiation pattern, which is given
by [73]:
f (m, k r ) =
N
n=1
A(k r )e
imnγ
(9.9)
where A(k r )is a k r -dependent coefficient, which can be ignored since we are concerned with the azimuthal dependence contained in f(m).