9 Aperiodic Order in Nanoplasmonics
345
demonstrated structured light carrying multiple values of OAM in the far field scattering region of arrays of metallic nanoparticles, in excellent agreement with analytical
calculations [77].
The unique features of Vogel spirals provide exciting new opportunities for the
engineering of OAM states using aperiodic nanostructures for a number of emerging
engineering applications in singular optics, secure communication, optical cryptography, and optical sensing.
9.2.5 Aperiodic Order and Number Theory
Numerical sequences and geometric patterns with deterministic, though apparently
haphazard behavior, have been deeply investigated in discrete mathematics and information theory [78, 79]. In particular, deterministic structures generated by numbertheoretic numerical sequences with flat-Fourier spectra have found numerous technological applications in different research areas ranging from the engineering of
acoustic diffusers to radar abatement (stealth surfaces), spread spectrum communication (jamming countermeasures, secure channel sharing), and the design of minimum redundancy antenna arrays (surveillance) in the RF regime [10]. However,
aperiodicity in number theory is still largely unexplored in the domain of optical
technologies.
Number theory is primarily concerned with the properties of integer numbers, but
encompasses a large spectrum of advanced techniques from virtually all branches
of mathematics [11, 12]. As a result, number theory provides numerous insights
and algorithmic approaches for the generation of different types of pseudo-random
systems, point sets, and aperiodic tilings with various degree of structural complexity
and Fourier spectral properties [10, 12].
Figure 9.7 shows representative examples of aperiodic particle arrays deterministically generated based on number-theoretic concepts in the real and complex
fields. In particular, we show a Gaussian prime array (Fig. 9.7a), a co-prime array
(Fig. 9.7b), and a Galois field array (Fig. 9.7c), along with the calculated Fourier
spectra (Fig. 9.7d–f).
Gaussian primes (GP) are Gaussian integers that are prime in the complex field,
and are defined by n +im, where n and m are integers and i is the imaginary unit [10].
We notice that primes of the form 4k − 1 in the ring of conventional integer numbers
are still primes in the complex field, but 2 and primes of the form 4k+1 can be factored
in the complex field (e.g., 2 = (1 + i)(1 − i), 5 = (2 + i)(2 − i), etc.). Moreover, if
x is a Gaussian prime, the four numbers ±x, ±i x are called the associates of x and
are also Gaussian primes. As a result, Gaussian primes are symmetric about the real
and imaginary axes.
By plotting the real and imaginary components of GP numbers as horizontal
and vertical coordinates, we can represent Gaussian primes geometrically in the
complex plane, and produce the highly symmetric pattern displayed in Fig. 9.7a.
However, despite the remarkable symmetry of the pattern, GP arrays are non-periodic.
345
demonstrated structured light carrying multiple values of OAM in the far field scattering region of arrays of metallic nanoparticles, in excellent agreement with analytical
calculations [77].
The unique features of Vogel spirals provide exciting new opportunities for the
engineering of OAM states using aperiodic nanostructures for a number of emerging
engineering applications in singular optics, secure communication, optical cryptography, and optical sensing.
9.2.5 Aperiodic Order and Number Theory
Numerical sequences and geometric patterns with deterministic, though apparently
haphazard behavior, have been deeply investigated in discrete mathematics and information theory [78, 79]. In particular, deterministic structures generated by numbertheoretic numerical sequences with flat-Fourier spectra have found numerous technological applications in different research areas ranging from the engineering of
acoustic diffusers to radar abatement (stealth surfaces), spread spectrum communication (jamming countermeasures, secure channel sharing), and the design of minimum redundancy antenna arrays (surveillance) in the RF regime [10]. However,
aperiodicity in number theory is still largely unexplored in the domain of optical
technologies.
Number theory is primarily concerned with the properties of integer numbers, but
encompasses a large spectrum of advanced techniques from virtually all branches
of mathematics [11, 12]. As a result, number theory provides numerous insights
and algorithmic approaches for the generation of different types of pseudo-random
systems, point sets, and aperiodic tilings with various degree of structural complexity
and Fourier spectral properties [10, 12].
Figure 9.7 shows representative examples of aperiodic particle arrays deterministically generated based on number-theoretic concepts in the real and complex
fields. In particular, we show a Gaussian prime array (Fig. 9.7a), a co-prime array
(Fig. 9.7b), and a Galois field array (Fig. 9.7c), along with the calculated Fourier
spectra (Fig. 9.7d–f).
Gaussian primes (GP) are Gaussian integers that are prime in the complex field,
and are defined by n +im, where n and m are integers and i is the imaginary unit [10].
We notice that primes of the form 4k − 1 in the ring of conventional integer numbers
are still primes in the complex field, but 2 and primes of the form 4k+1 can be factored
in the complex field (e.g., 2 = (1 + i)(1 − i), 5 = (2 + i)(2 − i), etc.). Moreover, if
x is a Gaussian prime, the four numbers ±x, ±i x are called the associates of x and
are also Gaussian primes. As a result, Gaussian primes are symmetric about the real
and imaginary axes.
By plotting the real and imaginary components of GP numbers as horizontal
and vertical coordinates, we can represent Gaussian primes geometrically in the
complex plane, and produce the highly symmetric pattern displayed in Fig. 9.7a.
However, despite the remarkable symmetry of the pattern, GP arrays are non-periodic.