344
L. D. Negro et al.
Fig. 9.6 a, b Analytically calculated far field radiation patterns of GA and μ-spiral with 2,000
particles at a wavelength of 633 nm for structures with a 0 = 14.5 μm. The far field radiation
patterns have been truncated with an angular aperture of 4 ◦ and 3 ◦ , respectively. c, d Fourier–
Hankel transforms of far field scattered radiation by GA and μ-spiral, respectively, summed over
the radial wavenumber k r . The numbers in the figures indicate the azimuthal Bessel order of the
corresponding FHD peaks
We see from the result in Eq. 9.9 that when mγ is an integer, the N waves in the
equation will be exactly in phase to produce an OAM peak with azimuthal number
m. For an irrational angle γ, this condition will never be exactly satisfied. However,
any irrational γ can be approximated by an infinite continued expansion of rational
fractions, each approximately matching the integer condition for the mγ product.
Therefore, for spirals generated using an arbitrary irrational number γ, azimuthal
peaks of order m (i.e, Bessel order m) will appear in the FHD spectrum due to all the
denominators q n of the rational approximations (i.e., the convergents) of γ ≈ p n /q n ,
resulting in a sequence of discrete azimuthal peaks in the OAM of scattered light
simply determined by the aperiodic geometry.
Using phase delayed interferometric measurements, we have recently recovered
the complex electric field distribution of scattered radiation by Vogel spirals and
L. D. Negro et al.
Fig. 9.6 a, b Analytically calculated far field radiation patterns of GA and μ-spiral with 2,000
particles at a wavelength of 633 nm for structures with a 0 = 14.5 μm. The far field radiation
patterns have been truncated with an angular aperture of 4 ◦ and 3 ◦ , respectively. c, d Fourier–
Hankel transforms of far field scattered radiation by GA and μ-spiral, respectively, summed over
the radial wavenumber k r . The numbers in the figures indicate the azimuthal Bessel order of the
corresponding FHD peaks
We see from the result in Eq. 9.9 that when mγ is an integer, the N waves in the
equation will be exactly in phase to produce an OAM peak with azimuthal number
m. For an irrational angle γ, this condition will never be exactly satisfied. However,
any irrational γ can be approximated by an infinite continued expansion of rational
fractions, each approximately matching the integer condition for the mγ product.
Therefore, for spirals generated using an arbitrary irrational number γ, azimuthal
peaks of order m (i.e, Bessel order m) will appear in the FHD spectrum due to all the
denominators q n of the rational approximations (i.e., the convergents) of γ ≈ p n /q n ,
resulting in a sequence of discrete azimuthal peaks in the OAM of scattered light
simply determined by the aperiodic geometry.
Using phase delayed interferometric measurements, we have recently recovered
the complex electric field distribution of scattered radiation by Vogel spirals and