Propagation of Ultrashort Optical Pulses in Fractal Objects

Abstract

In this paper, we study the features of the evolution of an electromagnetic pulse in fractal structures. Different fractal structures were considered, with different symmetry and different generators. Based on the electron dispersion law for fractal objects, an expression for the electric current density in the system under study was obtained. It was found that the fracton dimension does not significantly affect the dynamics of ultrashort optical pulses. Thus, ultrashort optical pulses do not feel the symmetry of fractal objects.

1. Introduction

Fractals are objects with fractional size [1] and self-similarity properties. They are often used to describe various phenomena and processes [2,3]. However, their more important application is the possibility of achieving new scientific results in various subject areas [4,5], mainly in physics [6]. Let us also provide some examples from nonlinear optics. These are plasmonic fractal structures that have been demonstrated in gigahertz applications [7]: the controlled generation of fractal light inside the laser cavity [8]; the production of compact active devices, such as fractal antennas, with a strong and broad spectrum response in the mid-visible/infrared range [9]; the development of waveguide structures based on fractals [10], which have great potential to increase the efficiency and sensitivity of photonic devices.

It is well-known that elastic oscillations are observed in fractal systems, whose local state is called a fracton [11]. Such states are objects of interest to modern physicists, as they can lead to new results in the fields of materials science [12], quantum physics [13] and gravity [14]. For example, we note a significant effect on the tunneling current, as evidenced by the results presented by [15].

On the other hand, a popular area of research in modern optics is the formation of short-duration localized electromagnetic pulses, including ultrashort optical pulses (USOP) [16,17,18,19,20,21], and the search for conditions under which they propagate stably Therefore, the aim of this work is to investigate the effect of the fracton size on the propagation dynamics of ultrashort optical pulses in a fractal medium.

The short duration of such pulses makes them especially attractive for optical-electronic systems and information transmission, since with a decrease in duration, the throughput and line speed increase. Another important and non-insignificant feature of extremely short optical pulses is their wide spectrum, which is extremely valuable in spectroscopic studies. Obtaining ultrashort optical pulses is the result of the implementation of the principle of compression—the focusing of optical radiation in time. The main points of focusing in time, by an analogy of the focusing of wave beams in space, are fast frequency or phase modulation and compression of the modulated pulse in a dispersive medium. If we are talking about the generation of pulses with a duration comparable to the period of optical oscillations of an electromagnetic wave, then the frequency scanning range should be definitely comparable to the carrier frequency. Currently, the most promising method for creating such fast modulation is self-phase modulation in a medium with practically inertialess electronic nonlinearity. An ideal compression system, by analogy with aberration-free focusing of a wave beam, assumes the implementation of frequency modulation that is linear in time and exact phase matching of the components of the broadened spectrum at the focal point. The practical implementation of ideal compression conditions is a relatively difficult task.

USOP can be generated directly from the laser due to the effect of self-induced transparency. These femtosecond lasers are quantum optical generators capable of generating laser pulses that contain a small number of electromagnetic field oscillations.

Since at present a pulse duration on the order of 50 picoseconds can be achieved with fairly affordable equipment using laser diodes, research in the field of creating pulses shorter than 1 picosecond has become topical. Currently, there are lasers capable of generating an ultrashort optical pulse with a duration of about 5 femtoseconds. There are reports about the creation of experimental systems with an attosecond pulse duration. One of the most effective options for obtaining ultrashort optical pulses is the compression of the modulated signal. Using self-modulation of a high-power laser pulse in a fiber light guide, which is a medium with Kerr-type inertia-free cubic nonlinearity, a frequency-modulated pulse is obtained, which is directed to an optical compensator consisting, for example, of two diffraction gratings and one mirror. Since the latter have anomalous dispersion, that is, long-wave radiation receives a greater delay than short-wave radiation, the conditions for their synchronization can, thus, be realized.

Two points should be noted. First, an ultrashort optical pulse has an extremely wide spectrum, both in the space and time domains. This means that it must sense changes in the structure of the substance being studied on many scales. Second, for fractals, changes in their structure and, therefore, key characteristics, such as dimension, are introduced into their generation by symmetry. So, for example, when we cut centered equilateral triangles from an equilateral triangle according to the standard procedure for constructing a Sierpinski carpet, a fractal structure with a Hausdorff dimension is obtained. However, when cutting the square from a square, the size will be different. Obviously, this is due to the symmetry of the original objects from which the fractal is constructed. So, it is very important to determine whether the USOP perceives the fracton size, i.e., the symmetry of the original subject on which the fractal is generated.

2. Power Laws of Dispersion

To simulate the dynamics of a pulse in a fractal medium, it is necessary to obtain the law of dispersion of the electrons in such a system. It is necessary to generalize the known electron dispersion laws for Fermi liquid, graphene-like materials to the case of fractional dimension. We assume that it should have a similar form: ε~p^α, p—the electron momentum.

Dispersion laws of such a type are often encountered in various physics problems. This law is particularly common in problems with Fermi and non-Fermi fluids, as well as problems with the interactions of these fluids with impurity systems.

Ideas derived from string theory, especially the idea of the ADS/CFT correspondence, have helped to make significant advances in the study of quantum critical phenomena [22,23]. Recent advances and certain achievements have been achieved, not least because well-developed technical methods in string theory have begun to be used in this field [24]. In the field of critical phenomena, since the 1970s, a clear understanding of the role of universality, scale and, thus, the ideas of the field theory has been attained [25,26], which predestined the success of this method. The close relationship of quantum theory with quantum important phenomena leads to a clearer understanding and mutual development of these fields.

It is well known that physical theories, as a rule, become consistent with a certain choice of parameters. The most famous example of this type is the appearance of consistent invariance near the phase transition when the scale of the critical oscillations becomes large and the correlation length tends to infinity. Similarly, many theories describing the properties of non-Fermi fluids are appropriate, and the Green function for them can be obtained from the ADS/CFT correspondence [26,27,28]. In this regard, of great interest are systems that lie “at the border” between Fermi fluids and non-Fermi fluids. The electronic states of such systems are called marginal Fermi fluids. Theoretically, this state has been predicted for quite some time, but it took a long time to find the documents that clearly show it. Meanwhile, comprehensive experimental studies of such materials are important as they can contribute to the development of new theoretical methods for their description. After all, if one could finally understand the behavior of marginal Fermi fluids, it would be possible to grasp much more complex non-Fermi fluid systems.

It was shown by [29] that a one- and two-dimensional ultrashort optical pulse can propagate stably in marginal Fermi fluids, arising as a result of the approximate equilibrium processes of dispersion and nonlinearity. We see possible applications of this effect in the basic ability to control the horizontal width of an extremely short optical pulse, which can be used in applications. In this case, the width of the limited short pulse will be determined by the length of its path in the non-Fermi fluid. We also note that the nature of the pulse propagation, especially depending on the value of v_k(k = k_F), is essentially dependent on whether the wavefront is flat or not. This allows one to experimentally verify the consequences of this method when studying the propagation of pulses in non-Fermi fluids with different wavefront curvatures.

In turn, this increases the interest in the description of nonlinear optical phenomena in Fermi fluids. Therefore, the dynamics of ultrashort optical pulses with plane wave fronts in Fermi fluids have been investigated recently [30]. In addition to the problem of plane wavefront stability, there are still a number of problems involved in going beyond one-dimensional approximation and studying the dynamics of optical propagation with permission for dispersion effects. The study of the kinematics of a local three-dimensional optical pulse on all spatial coordinates is equally interesting. This confirms the fact that the study of such pulses, called “light bullets” [31,32], has recently become very popular, mainly due to a large number of practical applications.

The approach developed below in the work corresponds to the ideology of the so-called “semi-holography” [33], when the dispersion law of a strongly interacting subsystem is made based on the correspondence of ADS/CFT and is further considered “classical” based on developed approaches based on the equations of the quantum theory of solids. It was shown by [33] that this approach has great advantages: the universal low-energy characteristics that are relevant to real systems are preserved, and the inclusion of a spatial lattice and impurities in the consideration is ensured.

The above dependence is typical and, thus, leads to the steady propagation of local pulses in two/three dimensions, commonly referred to as “light bullets” in the literature, which can occur in Fermi liquids. Calculations have shown that although there is diffraction propagation of the pulse in the direction opposite to that of propagation, in general, the pulse retains its shape. It should also be noted that there is partial curvature of the pulse front, which occurs due to diffraction.

At the same time, researchers’ attention is attracted by the Rashba effect, which is best known as the spin-orbit separation of two- and one-dimensional spectra in asymmetric quantum wells [34,35,36], underpinning the idea of creating one of the most promising spin-electronic devices—the spin field effect transistor [37], making the study of this effect increasingly attractive to modern researchers, both from a practical and theoretical point of view.

Based on this, the question arises regarding the effect of the Rashba interaction on the dynamics of ultrashort Fermi fluid pulses, which makes it possible to perform the spectroscopy of such an interaction, and the systems in this equipment can be used for optical information processing. Moreover, the proposed approach corresponds to the ideology of “semi-holography” [33] when the dispersion law is implemented based on the ADS/CFT correspondence.

Numerical calculations have shown that in the 2D case, stable nonlinear waves can occur and light pulses are localized in two directions, similar to “light bullets”. When a “bullet” propagates in the Fermi fluid, its propagation in the horizontal direction is rather weak, while the energy is mainly concentrated in the central region of the pulse [38].

Another example of a medium with such a dispersion law is graphene-like structures, such as silicene and germanene, which consist of a layer of silicon or germanium atoms in a hexagonal lattice [39,40]. A distinctive feature of these materials is that the orbital interaction is stronger than that of graphene. One of the most interesting predictions for them is the appearance of a band gap, which can lead to the appearance of a transition between the band and the topological insulator. At the same time, one cannot ignore the fact that silicon is still a key component of modern microelectronic devices.

In recent works, the authors have studied the problem of the propagation of ultrashort DC pulses in silicene waveguides [41]. It has been shown that starting from a certain time, an inverted signal is observed, while the amplitude of the inverted signal is almost twice the amplitude of the original signal. Thus, we can talk about the amplification of extremely short pulses with a marked change in their shape.

In silicene, since the silicon atoms are not completely in the same plane but are above and below it, the width of the gap can be controlled by applying an electric field in a direction perpendicular to the plane. In this case, due to the potential difference between the sublattices, the size of the gap is directly dependent on the applied constant electric field. It is clear that in this case, the applied random electric field as described above will play an important role. Random fields can arise both from charged impurities embedded between the planes of silicene and, for example, of an extrinsic nature. The electric field pulse will propagate in a medium with a random band gap, and the previous conclusions need to be clarified.

Based on the above scenarios, it is important to study the interaction of ultrashort pulses with silicene in a situation with a random electric field where new effects with many practical applications are expected.

Therefore, the introduction of electric field averaging provides a balance between the dispersion and nonlinearity of the system, resulting in amplitude conservation and pulse stability [42].

In a study by [43], the propagation properties of two- and three-dimensional ultrashort optical pulses in germanene, due to strong spin-orbit interactions, were investigated. It was shown that the number of electric field oscillations has a significant effect on the shape of an ultrashort optical pulse. It was found that in the case of both one and two oscillations of the electric field, the pulse propagates steadily without forming any “tails” behind it. The influence of the rotation–orbit interaction strength on the propagation of the ultrashort pulse is shown in the stability of the pulse.

There are also substances whose dispersion laws consist of two terms with different α. These are topological insulators.

The first mention of topological insulators can clearly be linked to work on the quantum Hall effect in a confined two-dimensional electron gas. If far from the boundary of the sample, the electrons in the case of the quantum Hall effect move in a limited region of space and, therefore, the sample does not conduct electricity. If the electrons are near the boundary, due to the “reflection” of the electrons from the boundaries of the sample, they can move infinitely and, thus, flow. In classical terms, in the volume of the sample, electrons move in the magnetic field in closed orbits and the sample is an insulator, while on the surface, electrons can be reflected off boundaries and current can flow. Note that the direction of the current is determined by both the spin of the electrons and the direction of the magnetic field. Further study of this effect took place regarding replacing the interaction with the external magnetic field with the rotation–orbital interaction, which in fact led to the discovery of topological insulators. It should be noted that the problem of the interaction with an intense external electromagnetic field, for example with the field of an extremely short optical pulse, is of great interest [44].

The Hamiltonian for a thin film of a topological insulator in the long-wavelength approximation can be written as:

$$H=(p_x^2+p_y^2)/2m+v_f(p_x{\sigma }_y-p_y{\sigma }_x)$$

(1)

Note that this efficient Hamiltonian generation for a Hamiltonian-based thin film for a bulk sample is given in several papers. Here p_x and p_y are the electron momentum components, m is the effective electron mass, σ_x and σ_y are the spin matrices and v_f is the Fermi velocity. Typical values of the Hamiltonian parameters, for example, for $B{i}_{2}T{e}_3$ are m~35 eV⁻¹ Å⁻², v_f~5 × 10⁻⁴ eV⁻¹ Å⁻². The Hamiltonian Equation (1) is easily diagonalized and sets the electron spectrum:

$$\epsilon (p_x,p_y)=(p_x^2+p_y^2)/2m+v_f\sqrt{p_x^2+p_y^2}$$

(2)

It was shown by [44] that extremely short optical pulses can be reliably transmitted in topologically insulating films. This effect could be useful in developing hybrid devices based on the interaction of light with electrons in topological insulators. We also note that the effect associated with forming a “tail” behind an extremely short pulse can be used to generate pulses in the terahertz range.

The authors also investigated the dynamics of ultrashort pulses propagating in thin films of topological insulators capable of exhibiting dispersion properties in a non-magnetic medium. An efficiency equation was obtained for the dynamics of ultrashort optical pulses in a thin film of topological insulators, taking into account the dispersion of the medium. It has been found that the shape of the ultrashort optical pulses is influenced by the value of the initial pulse amplitude, which is determined by the nonlinearity of the medium through which the pulse propagates.

Another important practical case is the study of the dynamics of ultrashort optical pulses propagating in thin films of topological insulators on boron nitride substrates. Note that such a choice of substrate is due to the ionic nature of the interatomic bonds, which results in the absence of “dangled” covalent bonds and charge trapping on the boron nitride surface [45]. In addition, it is much simpler and more convenient, including for practical purposes, to study thin films of topological insulators on substrates than in the free form. The stabilizing effect of the substrate on the pulse is revealed, which manifests itself in the absence of a “tail” behind the main pulse.

It is worth noting the growing interest in nonlinear light transmission in discrete waveguide structures. This is connected both with the practical usability of nonlinear optical effects and the fact that the propagation of light beams in such structures is similar to the motion of an electron in lattice crystals. The physical phenomena underlying this effect are also observed in other systems, such as semiconductor superlattices, biomolecular structures, Bose-Einstein condensate with cyclic potential, etc.

The unique properties of topological insulators, their definite similarity with graphene and the operation on the propagation of ultrashort pulses in graphene systems provide another impetus to study the problem of electromagnetic pulse propagation through a system consisting of several topological insulators. The results obtained earlier allow us to expect to discover new interesting effects in these structures, including solitons. The theoretical proof of the nonlinear localization of light in the periodic structures of coupled optical waveguides was first provided by [46]. However, there was only experimental confirmation for the existence of such spatially localized states as discrete solitons ten years later in gallium arsenide (GaAs)-based waveguide gratings. The authors studied the propagation of discrete solitons in a system of topological insulator thin films that act as waveguides. A model describing such propagation was built. It turns out that pulse propagation is significantly affected by the width of the initial pulse. When an electromagnetic pulse propagates in such a system of waveguides, the pulse is reversed from the original pulse and its amplitude increases; therefore, topological insulators can be used in devices to invert, as well as amplify, signals.

An important issue is taking into account the influence of the external electric field, which can have a significant impact on the dynamics of AC field pulses in a thin film of topological insulators. Note that the introduction of a constant electric field often brings about fundamentally new physical effects, since it establishes a preferred direction and reduces the symmetry of the system. It is, therefore, well known that in a medium with cubic nonlinearity, the introduction of a constant electric field can produce uniform harmonics, the production of which is initially forbidden due to symmetry. From the point of view of this method, it is clear that the introduction of a constant electric field can lead to an abundance of the pulse spectrum and, therefore, a change in its width, which is extremely important for practical applications. Also important in practical terms is the effect of a constant field on the pulse stability. It was shown by [47] that an applied constant electric field would prevent an increase in the duration of an alternating electric field pulse and “prevent” its expansion due to dispersion.

3. Model and Methods

Let us consider an ultrashort optical pulse with its wave vector directed perpendicular to the planes of the fractal structure (Figure 1).

The electron dispersion law for a fractal object can be written as:

$${\epsilon }_1\left(p_y,p_z\right)=V_F{\left(p_y^2+p_z^2\right)}^{0.5\sigma }$$

(3)

here (p_y, p_z) are the quasi-momentum components of the electron, V_F is similar to the Fermi velocity for fractals, σ is the dimension used to describe fracton states. It should be noted that we call it the fracton dimension, as described by [48]:

$$\sigma =\frac{2d_f}{d_w}$$

(4)

d_f is the fractal size of an object, d_w is the diffusion index.

Note that by fractal size, we mean the extent of space-filling or a measure of how geometrically irregular an object is.

For lattices with defects when there are some bonds and without them, the value of σ is considered analytically and coincides with the exponent in the electron dispersion law (3). It should be noted here that, due to the high symmetry, it is possible to calculate the exponent σ for some lattices. So, the symmetry of the original object and the symmetry of the “cut” regions determine the fracton size, and USOP propagate in such a medium. Note that below, putting the dispersion law in the form Equation (3) helps to calculate the analytical expression for the current.

The wave equation for the nonzero component of the vector potential of the electric field in the three-dimensional case can be written as:

$$\frac{{\partial }^{2}A}{\partial x^2}+\frac{{\partial }^{2}A}{\partial y^2}+\frac{{\partial }^{2}A}{\partial z^2}-\frac{1}{c^2}\frac{{\partial }^{2}A}{\partial t^2}+\frac{4\pi }{c}j=0$$

(5)

Since we are working at approximately low temperatures, when only a small region in the momentum space near the Fermi level contributes to the current, we can write the expression for the current density:

$$j=e{\displaystyle \underset{-\Delta }{\overset{\Delta }{\int }}{\displaystyle \underset{-\Delta }{\overset{\Delta }{\int }}v\left(p_y,p_z-\frac{eA}{c}\right)}} d{p}_{z}d{p}_y$$

(6)

here the momentum integral region is found from the condition of being equal in the number of particles in it and the first Brillouin region, $v\left(p_y,p_z\right)=\partial \epsilon \left(p_y,p_z\right)/\partial p_z$—electronic speed.

Next, we perform the conversion to a cylindrical coordinate system, ignoring the derivative with respect to the angle ∂/∂φ → 0 due to the small [49] charge-accumulation effect, allowing us to rewrite Equation (5) into the form:

$$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial A}{\partial r}\right)+\frac{{\partial }^{2}A}{\partial z^2}-\frac{1}{c^2}\frac{{\partial }^{2}A}{\partial t^2}+\frac{4\pi }{c}j=0$$

(7)

4. Simulation Results and Discussion

Research Equation (7) has been solved numerically [50]. The initial value is chosen in the form of a Gaussian pulse and can be written in the following form:

$$\begin{array}{l}A\left(r,z\right)=Q\cdot \exp \left(-\frac{z^2}{{\gamma }_z^2}\right)\cdot \exp \left(-\frac{r^2}{{\gamma }_r^2}\right),\\ \frac{dA\left(r,z\right)}{dt}=\frac{2uQ}{{\gamma }_z^2}\cdot \exp \left(-\frac{z^2}{{\gamma }_z^2}\right)\cdot \exp \left(-\frac{r^2}{{\gamma }_r^2}\right)\end{array}$$

(8)

where Q is the pulse amplitude at t = 0, γ_i is its initial width in the direction i = r, z and u is the initial speed in the axial direction z.

The development of an ultrashort optical pulse during its propagation in fractal-structured objects is shown in Figure 2.

To be sure, we take a Sierpinski carpet fractal with a generator (7,3) and fracton size σ = 1801 [51]. As in the case of integer-sized objects, the propagating pulse is quite stable, keeping the maximum energy in its central part. A localized electromagnetic wave packet inevitably propagates both in space and time under the simultaneous actions of dispersion and diffraction, which are present in any carrier. Significant research activity has been devoted to developing new ways to overcome these broadening effects in order to create stable localized wave packets. Such packets of a traveling wave, which are localized while retaining their spatiotemporal shape despite diffraction and dispersion effects, are usually called light bullets. As can be seen from the evolution of the propagation of three-dimensional ultrashort optical pulses shown in Figure 2, the light bullet changes its configuration somewhat, and the change in shape inevitably occurs over time due to the dispersion effects of the medium.

The effect of the fracton size on the dynamics of an ultrashort pulse is shown in Figure 3.

As can be seen from Figure 3, the fracton size and, thus, the shape of the fractal do not significantly affect the shape or intensity of the pulse.

5. Conclusions

In this paper, we propose a model that describes the interaction of ultrashort optical pulses with a medium containing fractal objects in the continuous medium approximation. The electron dispersion law is generalized to the case of the fractional dimension. Based on Maxwell’s equations, an effective equation that describes the evolution of an electromagnetic pulse is obtained.

As a result of the study, it was shown that a change in the fractal dimension and, hence, the type of the fractal and its symmetry does not significantly affect the dynamics of the USOP. This property paves the way for the use of different fractal structures for the development of optical devices. In addition, the analysis of the spatial and energy characteristics of the pulse showed that it propagates in structures with a fractal geometry while maintaining its localization area. This is an important result from the point of view of the possibility of using materials with a fractal structure to ensure the stable propagation of ultrashort optical pulses, for example, in fiber-optic lines.