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Anderson localization is an interference effect yielding a drastic reduction of diffusion—including complete hindrance—of wave packets such as sound, electromagnetic waves, and particle wave functions in the presence of strong disorder. In optics, this effect has been observed and demonstrated unquestionably only in dimensionally reduced systems. In particular, transverse localization (TL) occurs in optical fibers, which are disordered orthogonal to and translationally invariant along the propagation direction. The resonant and tube-shaped localized states act as micro-fiber-like single-mode transmission channels. Since the proposal of the first TL models in the early eighties, the fabrication technology and experimental probing techniques took giant steps forwards: TL has been observed in photo-refractive crystals, in plastic optical fibers, and also in glassy platforms, while employing direct laser writing is now possible to tailor and “design” disorder. This review covers all these aspects that are today making TL closer to applications such as quantum communication or image transport. We first discuss nonlinear optical phenomena in the TL regime, enabling steering of optical communication channels. We further report on an experiment testing the traditional, approximate way of introducing disorder into Maxwell’s equations for the description of TL. We find that it does not agree with our findings for the average localization length. We present a new theory, which does not involve an approximation and which agrees with our findings. Finally, we report on some quantum aspects, showing how a single-photon state can be localized in some of its inner degrees of freedom and how quantum phenomena can be employed to secure a quantum communication channel.

Transverse localization (TL) is found in media in which the refractive index is randomly modulated only orthogonally to the direction of propagation. In these paraxial systems, Anderson localization (AL) sustains nondiffracting beams: confined light tubes showing many potential applications including fiber optics, quantum communication, and endoscopic imaging. In this review we will summarize recent advances in disordered optical fibers, in which confinement is obtained thanks to localization, discussing the advantages with respect to standard fibers. First we will report the latest experimental results on transverse Anderson localization: the migration of localized states due to nonlinearity, self-focusing, wavefront shaping in the localized regime, and the single-mode transport in disordered paraxial structures. This last result is particularly important as it bridges the physics of Anderson localization to the single-mode properties of optical fibers.

Then we will show how the traditional description of Anderson localization, which was based on the analogy to electrons in a random potential, turned out to be in error and led to the prediction of a localization length depending strongly on the wavelength of the light, which was not observed. We also report on the alternative correct theory, which relies on an analogy to acoustical waves in the presence of random elastic moduli. Regarding quantum aspects, we will report on how a single-photon state localized in some of its inner degrees of freedom could be an effective resource in quantum communication and cryptography, increasing both the amount of information loaded per single particle and the security and performance of protocols based on localized photon quanta. Finally, we will review the so-called random quantum walks in which the dynamics of a single particle moving on a lattice conditionally to the state of an ancillary degree of freedom display localization under certain conditions. A further aspect of AL of quantum particles is the behavior of the multiparticle interference and of the particle statistics in quantum walks. In the first proof-of-principle photonic experiments, AL has been observed in the two-photon wave function. In this scenario, it could be possible to simulate even the fermionic statistics by proper manipulation of two-photon entangled states generated by single-photon sources.

In the last decades, the idea that Anderson localization could be applied to electromagnetic waves [

We start from the Helmholtz equation for the scalar field _{0} = _{0}
_{0} is the average refractive index in the disordered medium (disordered fiber), and λ is the wavelength in the medium.

By introducing a transverse Laplace operator

_{0}.

The first papers on transverse localization were focused on theoretical modeling and numerical simulations. The experimental realization of the effect required more than a decade from the appearance of the paper of De Raedt et al. [

The first experimental observation of TL (and actually one of the most unequivocal experimental manifestation of light localization) has been reported by Schwartz and coworkers [^{–4}) and a large disorder grain size (∼ 10 µm), thus the expected mean free path _{⊥}. The latter is the projection of the wavevector onto the _{⊥} = _{0} sin

Sketch of transverse localization: sketch of the scattering structure and illumination for the realization of the TL. Light, in the form of a plane wave defined by the wavevector _{
⊥
} and the projection of the wavevector on the propagation direction _{
z
}) are reported, together with the spectral parameter _{1}. To work as “paraxial defects”, the defect structures with index _{1} should be precisely parallel to

The big advantage of the optical induction is the possibility to completely rearrange the refractive index distribution, with a simple and fast rewriting procedure. The possibility to perform experiments with several realizations of the disordered

In 2012, Arash Mafi and coworkers [

By this procedure, homogenous fibers were realized with a realization of transverse disorder in the refractive index. The binary fiber approach provides several advantages: 1) the disordered refractive index distribution is permanent, 2) the refractive index mismatch between the two materials (Δ

In-fiber implementation of the Anderson localization enables the propagation of localized beams with the transverse size comparable to that of cores of commercial single mode optical fibers. Thus a single disordered fiber with sufficient transverse extension can act as a coherent fiber bundle [

The traditional theoretical description of Anderson localization of light, and, in particular, transverse Anderson localization [_{0} = 2

We call the traditional approach according to _{0}; see the dashed line in

1) In deriving the wave,

Probing the wavelength dependence of the average localization length.

Approximately neglecting this term leads to the strong dependence of the localization length on _{0}, and, hence, on the laser wavelength (dashed line in

2) An alternative wave equation has been derived by the authors of [^{2}(

There is a relevant debate about the fact that nonlinearity [

Localized states and nonlinearity. Panel

This self-focusing results from the peculiar interaction between disorder and thermal nonlinearity. In general, the refractive index of a nonlinear optical material, varies with the optical intensity _{0} + Δ_{2}
_{2} represent the coefficient of the nonlinear perturbation. A positive _{2} coefficient results in a converging wave front that can potentially surpass the diffraction limit. Conversely, a negative value of _{2} produces a defocusing nonlinearity, thus the expansion of the beam. In plastic binary fibers, one expects the slow thermal nonlinearity to yield a negative _{2}, thus defocusing. However, experimental measurements report instead a focusing nonlinearity. This unexpected effect is explained as follows [

This effect makes a local and optical tunability of the localization length possible, enabling to drive the position of the localized states in a form of localization-mediated beam steering. The steering effect is reported in

Light steering in the localization regime. Panel

Nonlocality obviously works also when more than two modes are involved. The behavior of a group of localized modes is visualized in

The Anderson localization (AL) scenario typically comprises a disordered system supporting states which are strongly localized at different locations in space and at different energies [

While the majority of the studies on AL are focused on the measurement of transport-related quantities (such as diffusion or conductance [

In the case of disordered optical fibers, one may ask, to which extent a localized state operates as a single mode hosted in the core of single-mode fibers. This issue has been extensively studied in [

In contrast to multimode fibers, disordered binary fibers (DBF) show peculiar transmittance maps. The transmittance map is the total (integrated over the whole fiber tip output) intensity measured as a function of the injection location, and measured with the setup shown in

Probing single mode nature of localized states. The sketch reports a scheme of the experimental setup. Legend: CW laser, continuous wave laser; M, mirror; BS, beam-splitter; OBJ, objective; DBF, disordered binary fiber. Panel

Now it is interesting to further investigate the nature of these transmittance hotspots. The most accessible feature is the intensity profile measured at the fiber output: this is reported at _{
i
} _{
i
}, _{
i
}, _{
i
}) in

Mode fingerprints. Each panel reports the spatial profile of the intensity found for the correspondent location in

To verify this picture, one should observe where the mode’s fingerprint is found while scanning the input of the fiber. To perform this measurement systematically, the authors engineered a specialized observable that is the degree of similarity _{1}, _{2})_{
i
}, it produces a Q-map _{
a
}, _{2}) = _{
a
})_{2})_{
a
} corresponds to a location of higher intensity of the mode _{
i
}. By computing _{
a
}, _{2}) for all _{2} in the area of view, we retrieve the Q-map reported in

Q-maps. Panels

A similar situation is found in _{
i
}. The two modes are only barely overlapping: energy is not flowing from one localized state to the other. The dark area in both maps corresponds to locations in which no intensity is transferred to the localized state. Note that the small displacements of the input inside the dwelling area do not cause any modification in the mode fingerprint. Multimode light structures would give rise to a pronounced flickering of the image due to the difference in phase delay of the different modes. The absence of such flickering is a relevant proof of the single-mode nature of the light structures supported by the DBFs.

The fact that the dwelling areas of different modes are barely overlapping is consistent with the picture in which localized states are orthonormal. A definitive confirmation of orthonormality requires to measure all the relevant observables: dwelling area, fingerprint, spectral parameter, and polarization. Such a challenging experiment (requiring the full characterization of thousands of modes) has not yet been carried on to our knowledge.

On the other hand, _{
i
}), provides a

Regarding polarization, [

The summary is as follows

1) High transmission channels in a fiber are sparse.

2) They are separated by a barely transmitting “sea”.

3) Independently on how (and where) light is coupled to a fiber, each transmission channel retains its fingerprint (output profile).

4) Modes are excited in alternative fashion (i.e., the same input location activates only a transmission channel at time).

In other words, localized states of a disordered binary fiber behave exactly like the single modes of conventional single mode fibers showing the same property: namely the “resilience to the launch conditions.”

Disorder binary fibers (DBF) are a unique architecture [

On the other hand, the concept of designed disorder [

In order to implant “designed disorder” into glassy optical fibers, the authors of [

In order to transfer the advantage of DBF to glasses, Gianfrate and coworkers [_{
x
} and _{
y
} are integer numbers between 0 and ^{2} defects (square with side _{
Mx
} along _{
My
} along

Localization length in direct laser written disorder. Localization length

The realization of localization induced by direct laser-written defects is the first step towards a new generation of glass-based optical fibers characterized by low absorption and greater stability with respect to their plastic counterpart. The ability of tuning the defect position will open the possibility to test the concepts of designed disorder directly in optical fibers, thus paving the way towards unprecedented applications.

Since the appearance of Anderson’s seminal 1958 article [

The rather ad-hoc scaling argument of the gang of four has been given two complementary field-theoretic fundaments: the self-consistent localization theory of Vollhardt and Wölfle [

1) Calculating a “unrenormalized” (or “reference”) conductance _{0} from the disorder statistics of the spatially fluctuating potential

2) Applying the

- scaling equations (scaling theory/nonlinear sigma model)

- self-consistency relation (self-consistent localization theory)

- potential-well relation.

It is interesting to note that as the potential-well analogy is not based on the assumption of weak disorder (i.e., the assumption that the relative variance of the potential fluctuations is small), one can apply the standard effective-medium theory for strong disorder, namely the coherent-potential approximation (CPA) for calculating _{0} [

The standard (Anderson) model for electron localization is given by the following Hamiltonian on a simple hypercubic lattice [_{
i
} = < _{
i
} are assumed to be independent random variables, which are distributed according to a distribution density

In the continuum description [

As there is no difference from the mathematical standpoint between a time-Fourier transformed classical wave equation (Helmholtz equation) and the Schrödinger ^{2} with −^{2}, where

Considering acoustical waves in a disordered medium, the disorder may come from 1) spatial density fluctuations or 2) spatial fluctuations of elastic moduli. John et al. [

If the density _{0} + Δ

Now the term ^{2}Δ^{2} term was multiplied with the spatially varying permittivity

For later reference, let us call a stochastic wave equation with fluctuating coefficient of ^{2} a “potential-type” (PT) equation and, if the elastic modulus fluctuates, a “modulus-type” (MT) stochastic equation.

Pinski et al. [^{2}(

If a wave is experiencing disorder like electrons in an impure metal, the waves are repeatedly scattered. This multiple scattering process can be interpreted as a random walk, featuring a diffusion constant

The diffusion approximation assumes that after each scattering event, the phase memory is lost. However, if one follows the scattering amplitudes with phases _{
ij
} (where _{
ij
} is the distance between two adjacent scattering centers) in a frozen medium, the phase memory is in principle not lost. This has dramatic consequences for recurrent partial paths, i.e., paths with closed loops: the phase of the recurrent path is exactly equal to that going in the reverse direction (see

Visualization of two interfering scattering parts, one going clockwise along the loop and the other anticlockwise.

For describing the interference mechanism, Abrahams et al. [

For ^{
d−2}, where ^{−L/ξ
}, where

Sketch of the scaling function as anticipated by Abrahams, Anderson, Licciardello, and Ramakrishnan [

The scaling function (

In two dimensions, the scaling

The localization length is given by the value _{1} ≈ 1. The reference conductance _{0} is the diffusion-approximation conductance_{0} the correlation length _{
c
} of the disorder fluctuations (see below)

From

As indicated above, almost the entire literature on Anderson localization (AL) of light is based on the potential-type wave equation, i.e., a wave equation in which the spatially fluctuating permittivity

Maxwell’s equations in a medium with spatially varying permittivity

For deriving a wave equation for the electromagnetic fields, one can either solve the electrical field

The traditional procedure (potential-type approach, PT) was to solve

This equation is mathematically equivalent to a stationary Schrödinger equation for an electron in a frequency-dependent random potential

We now want to check the validity of the approximation made in

One can estimate the error made in

If one solves the Maxwell

In order to describe transverse Anderson localization, we first map this problem to a two-dimensional problem. We then use the paraxial approximation to map the

We now consider an optical fiber with transverse disorder, i.e., the permittivity exhibits spatial fluctuations in

Because our system is translation invariant with respect to the ^{2} of a true two-dimensional system.

For _{0} + _{
z
}) (_{0} − _{
z
}) ≈ − 2_{0} (_{
z
} − _{0}) ≡ −2_{0}Δ_{
z
}, which is called the _{
z
} refers to the Fourier component of the

Introducing a “time” _{0} (which has the dimension of a squared length), this equation acquires the form of a Schrödinger equation of an electron in a medium with a randomly varying effective mass:

As stated above, such a model is related with a stochastic tight-binding model with a spatially varying hopping amplitude (“off-diagonal disorder”) [

Let us now compare the MT

By comparing _{0} on the localization properties: in the exact MT _{0} enters only as a prefactor of the _{0} does not enter at all. This is, however, completely different for the _{0} is the prefactor of the fluctuating-disorder term, which governs the mean-free path and hence the localization length.

How come that the predictions of two wave equations which are supposed to describe the same physical situation, namely, the wave propagation (or localization) of samples with transverse disorder, differ in such a drastic way? The difference can be traced back to the fact that in deriving (

The most important aspect of Anderson localization is the disordered environment. In the case of electromagnetic wave, the disorder may appear as randomly distributed scatterers or a spatially fluctuating permittivity ^{2}(

Lord Rayleigh considered in his seminal papers on the blue color of the sky [^{4} law for the scattering cross-section, which is inversely proportional to the mean-free path. It is known nowadays that this law becomes ^{
d+1} in

Because in our effective two-dimensional system the wave spectral parameter ^{2} is replaced by

We shall demonstrate in

For weak disorder^{1}
^{2}〉, and the correlation length _{
c
} of the fluctuating inverse permittivity (modulus). Here Δ

The correlation length of the spatially fluctuating modulus is an important parameter, because it defines the characteristic length scale of these fluctuations. It is defined by the length scale of the spatial decay of their correlations:

In this introductory subsection, we consider a simplified MT Helmholtz equation for a schematic scalar wave function

The corresponding Green’s function obeys

It has been shown in [

We now represent the correlation function schematically by introducing an upper wavenumber cutoff _{0} is a dimensionless constant and

We fix _{0} by requiring that the _{0} = 4^{2} 〉 / 〈^{2}. The inverse mean-free path is given by the imaginary part of Σ, multiplied with _{⊥} (see the next subsection):

The integral in _{⊥} ≤ _{
c
}, so that we obtain from the Born approximation (

For higher values of the spectral parameter, the Born approximation is insufficient, and we need a nonperturbative approach. Using a hand-waving argument, stating that it is inconsistent to work with two different Green’s functions, one may replace _{0} (

The SCBA may be obtained more rigorously by field-theoretical techniques [

For our detailed calculations^{2}
_{
m
} the mass density, and _{Σ}(_{Σ}′(_{Σ}(

The SCBA self-consistent equation for Σ(

From the local Green’s function, we obtain the

For

Making a variable change ^{2} and neglecting the imaginary part of

We now want to relate Σ″(

This generalizes the Born-approximation result (

The SCBA for the PT approach, due to John et al. [

As in the MT case, we have for the mean-free path

As the vector character of the magnetic field enters into our mean-field treatment only by doubling the Green’s functions, we return to a scalar description of the field amplitude.

The multiple scattering of waves in a turbid medium can be well described in terms of a random walk along the possible paths among the scattering centers [^{3}

As a matter of fact, within the saddle-point approximation (SCBA) one is able to calculate the mean-field diffusion coefficient _{0}(_{0} = _{0}/_{
F
}, where _{0} is the Drude conductivity and _{
F
} the density of states at the Fermi level. _{0} is obtained by considering the Gaussian fluctuations of the field variable _{saddle}(

This diffusivity may be related to the dimensionless reference conductivity _{0} by the Einstein relation [

We see that _{0} and _{0} in our model are equal to each other within a factor of order unity. In two dimension, the conductivity is also equal to the conductance. This quantity is relevant to the scaling approach of Anderson localization, which will be explained in the beginning of the next section. For

Using the self-consistent localization theory [

So, for “times” _{0} smaller than _{
ξ
} the intensity diffuses regularly according to ^{2} (_{0}(_{
ξ
}(^{2} (^{2}(

We remind ourselves that _{0}(_{0}(

We have solved both for the MT and the PT cases of the SCBA _{0} = 2_{Σ}(_{0} =

We observe the following features:

• In the MT case, all four curves fall on top of each other (because _{0} enters only into the definition of

• In the PT case, one obtains four different curves.

• Furthermore, in the PT case, the curves enter into the negative

How can one estimate the average localization length from this calculation?

The distribution of the localization length is determined by the function _{min}, indicated by the arrow in _{min}. On the other hand, in the PT case there is a broad range of values for _{0}. So, the average value of _{0}, as demonstrated by the numerical calculations by Karbasi et al. [

Reference conductance, which is proportional to the logarithm of the localization length _{0}(_{0}. Broken lines: _{0} = 1 _{
c
} = 8 ^{−1} [

In view of the fact that in our measurement we did not find a dependence on _{0} and that the SCBA of John et al. for the PT approach [

We stated in the beginning that we experimentally verified [

For the intensity _{
i
} in the (_{
i
}). As can be seen from _{
i
}.

Let us consider now, in more detail, cases in which one may obtain a wavelength dependence of the localization length. In our experiment, reported in [_{min}, which is _{0} independent. On the other hand, if one would work with a narrow-aperture laser, eventually tilted by a certain angle _{⊥} = _{0} sin(_{0} as prefactor, certainly the mode one may pick up by this procedure will be a different one if _{0} is changed. This opens an interesting method for further investigation of the localized single modes.

In this section we have presented a comprehensive theory of transverse Anderson localization of light. We started to derive the appropriate stochastic Helmholtz equation for electromagnetic waves with spatially fluctuating permittivity. We have shown that the potential-type approach, which is analogous to the Schrödinger equation for an electron in a random potential with the potential depending on the spectral parameter

Within the modulus-type approach, the localization length, i.e., the radius of the transmitted modes, diverges as the spectral parameter (which is proportional to the square of the azimuthal angle between the direction of the incident radiation and the optical angle) vanishes. This must be so, because a ray in the direction of the optical axis does not experience transverse disorder. The potential-type approach, however, implies a finite mean-free path at zero spectral parameter, and the predicted spectrum penetrates into the negative range of

At the end of this section, we would like to comment on the possibility of observing localization of light in three-dimensional systems. As mentioned in the introduction, despite of intensive efforts, this has not been observed until now. We emphasized that the modulus-type theory is analogous to sound waves in solids with spatially fluctuating shear modulus. There it is known that localized states exist at the upper band edge, which in solids is the Debye frequency. In turbid media the analogue of the upper band edge is the inverse of the correlation length of the disorder fluctuations. So if it would be possible to prepare materials with spatial fluctuations of the dielectric modulus, which have a correlation length of the order of the light wavelength, we expect chances for observing 3-dimensional Anderson localization.

According to the seminal studies by Anderson regarding single-particle evolution in lattices, the disorder in the system leads to localization of the wave-function. As we have illustrated in the first sections, such a phenomenon is well explained by quantum mechanics in the case of electrons and by classical electrodynamics in the case of light in the classical limit; i.e., no quantum effects are involved. In particular, localization is the result of constructive and destructive interference among the multiple paths of the particle. Being an explicit example of the wave-like behavior of quantum particles, the observation of AL in single-photon states does not display any substantial difference with respect to the experiments carried out with classical light. However, single-photons are one of the most promising candidates for quantum information processing in the context of computation, simulation, and cryptography [

This section regarding quantum AL is organized as follows. First, we introduce the quantum walks model and present single-photon experiments in the context of AL. We further provide practical applications of localized single-photon states in quantum cryptography protocols. Second, we illustrate two-photon quantum walks experiments and the effect of particle statistics in the localization.

The concept of quantum walks (QW) was first formulated as a generalization of classical random walks (RW) [

The evolution operator in the discrete-time scenario is the combination of the coin and shift action, namely

The QWs evolution operator can be modified for different tasks. For example, the QWs paradigm has been exploited to observe topological-protected states [

There are further examples of single-photon localization regarding continuous-time QWs. They are typically realized exploiting continuous coupling among waveguides arranged in a lattice in photonic chips. In this scenario, the time coordinate is again replaced by the distance _{
d
} is the single-photon amplitude at the site _{
ij
} are the couplings among the modes of the lattice that are expressed in the Hamiltonian

Concerning all mentioned quantum experiments, it is worth noting that the localized single-photon distribution has the same properties as the distribution of localized modes of classical light described in

Quantum computing could undermine the security of some of the current cryptographic protocols. An example is given by the RSA protocol security which is based on the difficulty for a classical computer to find prime factors of large integers, while a quantum computer solves the same problem in polynomial time [

Single-photon localized states are examples of qudits, where the

Experimental apparatus for quantum cryptography using single-photon localized states. Alice encodes her qudits by preparing single-photon states via a spatial light modulator. She chooses between two bases, namely (K) the eigenstates of the multimode fiber that localize after the propagation and (X) the states that spread after the fiber. Bob measures in the K basis or in the X basis inserting one or two lens before the detection stage. After the comparison between the basis choices by Alice and Bob, they can extract a secure key.

Single-photon localized states do not add any further insight into AL with respect to experiments based on wave interference. Nevertheless, the proper description of quantum light is within the framework of second quantization. This representation is necessary for describing many-particle evolution. The electromagnetic field can be expressed by the boson annihilation _{
ij
} are the element of the QW evolution operator in the occupation number representation. One of the most famous examples of two-photon interference, the Hong-Ou Mandel (HOM) experiment [

Two-photon interference has been investigated in the regime of AL. The main result that emerges from these studies is that the way in which the system approaches localization strongly depends on its initial state. In _{1}, _{2}) defined as the probability to detect one photon in the position _{1} and the other in _{2}, averaged over different disorder configurations. For example, in the case two identical photons injected in the QW in positions 0 and 1 in the state _{1}, _{2}) has the following expression_{1} and _{2} respectively and |_{
in
}

Two-photon quantum localization. _{1}, _{2}) of a two-photon wave function in a one-dimensional Anderson lattice. The four scenarios report the distributions for different input states in an intermediate time evolution, where the two photons have the same chance to be localized or to spread ballistically. _{1}, _{2} ∈ [−4,4] and _{1}− _{2} = Δ for the four initial states.

In this section we have illustrated Anderson localization (AL) in the context of quantum light, presenting the most relevant results for what concerns the experimental realizations and applications. We have first formulated AL in the context of quantum walks (QW). We have then described the use of localized states in quantum cryptography. In the end we have illustrated the problem of localization in quantum optics by considering multiphoton states. Up to now the investigation of multiphoton AL localization was confined to the two-photon case. The reasons are various. It is still debated in the literature, whether the results reported in the quantum experiments can be reproduced by classical light, i.e., by wave interference. For instance, in [

The story understanding transverse localization of light in the last four decades has been one of constant advances. With respect to the first formulations (which reported just numerical evidences [

1) Photorefractive crystals [

2) Polymeric binary fibers have been proposed in 2012 and successively further improved. The fabrication technique for these items is extremely cheap and straightforward (if one has access to a fiber drawing tower) and enables realizing kilometer-long fibers starting from a few-centimeter-sized preform. In binary fibers the refractive index mismatch is 0.1 (employing PMMA and Polystyrene as plastic components of the preform), and the disorder can be obtained easily with a grain size of the order of a micron. The advantage of this approach is that the micrometric sized defects need not to be individually fabricated: it is the transverse thinning, affecting collectively all individual strands in the preform that produces this fine-scale disorder. Thus binary fibers support strong localization in the visible range. Binary fibers have been thus extremely successful in terms of potential applications. It has been demonstrated that they can support nonlinearity and switching, image transport, wave front shaping, controlled focusing, quantum communication and key distribution, and image transport. The drawback of binary fibers resides in the large losses: currently between 50 and 100 dB/m. These losses are exceeding the ones expected for the intrinsic scattering and absorption of the plastic material: this poor performance is probably due to the assembly and drawing stage (carried on in a unclean environment), which introduces microscopic dust in the preform. Due to these losses, all the experiments have been carried on longitudinally small pieces of binary fibers (few tens of centimeters).

3) Glass-based binary fibers have been fabricated since 2014 from a porous glass. A rod with initial diameter of 8 mm produces air-holes with diameter varying between 0.2 and 5 µm. This approach promises all the advantages of glass (lower losses and enhanced stability) together with easy fabrication. This potential has already been demonstrated in recent results including new applications such as localization-based random lasing. The only drawback of this approach is related to the nonhomogeneity of the disorder. Indeed, air holes tend to be located at the outer boundary of the fibers due to the fabrication process and, thus, eventually turning localized states into leaky modes.

4) Employing fiber drawing (both in the glassy or plastic versions), it is impossible to get a direct control on the position of the defects. This drawback has been circumvented in 2020, employing a direct laser writing approach. Direct laser writing is still prone to high losses due to inefficient coupling and small refractive index contrast. Nevertheless, by tuning individually the paraxial defect positions, it is possible to test how extensive localization properties depend on specific configurations. Thus direct laser written localization can be employed as a test-bench to find out how different localization properties are affected by varying the disorder configurations. Then, if performance enhancement is found, the optimal configurations can be translated to the more efficient fabrication approaches.

If the technological progress on these platforms continues just at the same rate as in the last years, we can envision that one (or perhaps more than one) of these platforms will find its way to the application and industrialization in the next few years.

ML reviewed the experiments carried out with coherent light. WS and GR wrote the theoretical background. TG reviewed the experiments with quantum light. All the authors discussed the structure of the work and contributed to the writing of the paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

ML acknowledges Fondazione “CON IL SUD,” grant “Brains2south,” Project “LOCALITIS” and “Regione Lazio, grant “Gruppi di ricerca 2020,” and Project “LOCALSCENT.”

The theory may be generalized to include strong disorder using the coherent-potential approximation [

The reader, not interested in these details, may continue with the next subsection.

It is important to note that the “diffusivity”