9 Aperiodic Order in Nanoplasmonics
347
broader spectrum of spatial frequencies that are associated to several characteristic
length scales. Interestingly, several peaks appear in Fourier space related to the most
frequent distances between prime numbers [10, 80].
The discussion above makes us appreciate clearly the irreducible complexity that
often arises from the simple arithmetic properties of integer numbers. To this regard,
we notice that most number-theoretic functions (e.g., co-primality, discrete logarithm, Euler’s ψ, Möbius function, Riemann’s zeta function, Dirichlet L-functions,
etc) display apparent random behavior despite the large number of symmetry properties, easily detected when constructing the corresponding geometrical arrays.
This aperiodic behavior, ubiquitous in number theory, is fundamentally related to
intractable arithmetic problems (e.g., distribution of prime numbers, factorization,
etc). The role of randomness in number theory has been deeply discussed in the
framework of Algorithmic Information Theory (AIT) [78, 79].
Pseudo-random particle arrays with constant Fourier spectra can be conveniently
designed based on the theory of finite Galois fields [10]. In abstract algebra, a field is
a set of elements with addition, subtraction, multiplication and division (except by 0)
operations that satisfy the usual commutative, associative, and distributive laws.
Galois fields, named after the French mathematician Évariste Galois, are fields
with a finite number of elements (i.e., finite order fields) and have found numerous
applications in physics, communication theory, error-correcting code, cryptography,
and even artistic design [10]. As a simple example, a residue system modulo a prime
p forms a finite field (i.e., a Galois field) of order p, which is indicated by GF(p).
Of particular importance are finite number fields of order equal to a prime power p m
(i.e., with pm elements), where p is a prime number and m is a positive integer. A
Galois field of order p m is usually denoted as GF(p m ).
In particular, Galois sequences derived from GF(2 m ) have unique correlation
properties and possess flat Fourier spectra but, in contrast to other pseudo-random
binary sequences (i.e., Legendre sequences), are efficiently generated by a linear
recursion [10, 81–83]. Galois nanoparticle arrays can be constructed by generalizing
Galois recursions in two spatial dimensions as detailed in Refs. [10, 84]. Figure 9.7c
shows a calculated Galois particle array. The reciprocal Fourier space is shown in
Fig. 9.7d, and features a broad distribution of spatial frequencies without any welldefined Bragg peak, similarly to the white spectrum of disordered random media.
Two-dimensional Galois arrays possess a high density of spatial frequencies,
theoretically a flat measure for infinite-size arrays. This property has been used to
improve the image resolution of X-ray sources in astronomy [10].
Aperiodic arrays of metal nanoparticles generated according to number-theoretic
functions have been explored only recently in the context of plasmonic scattering
and field localization for optical sensing device applications [80, 85, 86].
However, since number-theoretic methods mostly provide asymptotic results, their
applicability to finite-size aperiodic structures remain fundamentally limited, with
remarkable exceptions in cryptographic domains (e.g., pseudo-random generators,
optical cryptography).
347
broader spectrum of spatial frequencies that are associated to several characteristic
length scales. Interestingly, several peaks appear in Fourier space related to the most
frequent distances between prime numbers [10, 80].
The discussion above makes us appreciate clearly the irreducible complexity that
often arises from the simple arithmetic properties of integer numbers. To this regard,
we notice that most number-theoretic functions (e.g., co-primality, discrete logarithm, Euler’s ψ, Möbius function, Riemann’s zeta function, Dirichlet L-functions,
etc) display apparent random behavior despite the large number of symmetry properties, easily detected when constructing the corresponding geometrical arrays.
This aperiodic behavior, ubiquitous in number theory, is fundamentally related to
intractable arithmetic problems (e.g., distribution of prime numbers, factorization,
etc). The role of randomness in number theory has been deeply discussed in the
framework of Algorithmic Information Theory (AIT) [78, 79].
Pseudo-random particle arrays with constant Fourier spectra can be conveniently
designed based on the theory of finite Galois fields [10]. In abstract algebra, a field is
a set of elements with addition, subtraction, multiplication and division (except by 0)
operations that satisfy the usual commutative, associative, and distributive laws.
Galois fields, named after the French mathematician Évariste Galois, are fields
with a finite number of elements (i.e., finite order fields) and have found numerous
applications in physics, communication theory, error-correcting code, cryptography,
and even artistic design [10]. As a simple example, a residue system modulo a prime
p forms a finite field (i.e., a Galois field) of order p, which is indicated by GF(p).
Of particular importance are finite number fields of order equal to a prime power p m
(i.e., with pm elements), where p is a prime number and m is a positive integer. A
Galois field of order p m is usually denoted as GF(p m ).
In particular, Galois sequences derived from GF(2 m ) have unique correlation
properties and possess flat Fourier spectra but, in contrast to other pseudo-random
binary sequences (i.e., Legendre sequences), are efficiently generated by a linear
recursion [10, 81–83]. Galois nanoparticle arrays can be constructed by generalizing
Galois recursions in two spatial dimensions as detailed in Refs. [10, 84]. Figure 9.7c
shows a calculated Galois particle array. The reciprocal Fourier space is shown in
Fig. 9.7d, and features a broad distribution of spatial frequencies without any welldefined Bragg peak, similarly to the white spectrum of disordered random media.
Two-dimensional Galois arrays possess a high density of spatial frequencies,
theoretically a flat measure for infinite-size arrays. This property has been used to
improve the image resolution of X-ray sources in astronomy [10].
Aperiodic arrays of metal nanoparticles generated according to number-theoretic
functions have been explored only recently in the context of plasmonic scattering
and field localization for optical sensing device applications [80, 85, 86].
However, since number-theoretic methods mostly provide asymptotic results, their
applicability to finite-size aperiodic structures remain fundamentally limited, with
remarkable exceptions in cryptographic domains (e.g., pseudo-random generators,
optical cryptography).