# Extreme and Topological Dissipative Solitons with Structured Matter and Structured Light

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structuring of a Medium

#### 2.1. Molecular J-Aggregates

#### 2.1.1. Model of J-Aggregate and Governing Equations

**P**of the molecular system. This interaction is taken into account in the rotating wave approximation, and ${\mathsf{\mu}}^{12}$ is the dipole moment of the molecule for the transition $1\to 2$. In this form, the Hamiltonian of the system is time-dependent. The transition to a stationary Hamiltonian is carried out by an unitary transformation of system operators with the replacement of old operators ${B}_{m}$ and ${D}_{m}$ with new ones ${b}_{m}$ and ${d}_{m}$:

_{0}

#### 2.1.2. Governing Equations

_{me}〉. In order to close the system of equations into which it enters, it is necessary to carry out factorization of many-particle terms in these equations [23]. As a result, we find

#### 2.1.3. Bistability for Molecular J-Aggregates

#### 2.1.4. Stability of Homogeneous Distributions

#### 2.1.5. Discrete Switching Waves and Dissipative Molecular Solitons

_{N}consisting of N molecules of pseudo-isocyanine chloride (PIC:Cl) where the value N can reach hundreds and thousands. For moderate N, the aggregate stable geometry, energies and intensities of lowest singlet electronic transitions can be found by quantum-chemistry methods [32]. Here, as well in the previous Sections, we model J-aggregate with a linear or circular chain of N three-level molecules each of them interacting with laser radiation and with other molecules via radiation emitted by molecules. Two lowest electronic levels 1 and 2 form an optical transition in quasi-resonance with the laser radiation frequency ${\omega}_{0}$. The third level is introduced to describe annihilation of excitations on two neighboring molecules: One of them is deactivated, and the other is activated third level with subsequent relaxation to the second (2) or ground (1) state. The frequency of transition from the ground to the third state is close to double frequency of the main transition 1→2, i.e., ${\omega}_{31}\approx 2{\omega}_{21}$. The intermolecular distance a is much less than the radiation wavelength, ${\lambda}_{0}=2\pi c/{\omega}_{0}$, where c is the light speed in vacuum. The laser radiation is linearly polarized. The governing equations were introduced in the previous sections; here we use in simulations their simplified version neglecting terms corresponding to many-particle interactions.

#### 2.2. Organic Thin Films: Bistability and Switching Waves

## 3. Light Structuring

#### 3.1. Model of a Laser with Saturable Absorption and Governing Equations

#### 3.2. Topological Laser Solitons

**r**on these lines is parallel to

**S**: $[{\nabla}_{3}\mathbf{r}\times \mathbf{S}]=0$.

**S**. The length of the curve is measured from its arbitrary point. The amplitudes a and the full phase $\Phi $ are considered to be real, the latter being represented as the sum of the azimuthal and radial (depending on l) phases:

_{U}and closed ${N}_{C}$ vortex lines, and the torsion index of the closed lines $s$—fully characterize the topological structure of solitons presented in Figure 11. Alternatively, the structure may be characterized by phase incursions while circling the vortex lines and along them. These indices can also be connected with the traditional form of topological indices, including linking number [65].

**S**. A point

**r**

_{3}corresponds to energy source if div

**S**(

**r**

_{3}) > 0 and to energy sink if div

**S**(

**r**

_{3}) < 0. Surfaces div

**S**(

**r**

_{3}) = 0 separate domains of energy sources and energy sinks of toroidal shape, see Figure 12.

**S**). Lines of energy flow are determined by the system of three ordinary differential equations of the first order:

#### 3.3. Hysteresis

**M**and $\widehat{\mathbf{J}}$ are calculated with respect to the structure center

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The bistability of the stationary population of the second excited level with a change in the Rabi frequency without taking into account the many-particle corrections. The coefficient of exciton-exciton annihilation takes the values: $\alpha =0$, 1, 5, 10, 15, and 25; frequency detuning $\overline{\text{}\Delta}=-10.$

**Figure 2.**Bistable dependence of the population of the second (excited) level on the Rabi frequency for stationary homogeneous states.

**Figure 3.**Dependence of the width of the bistability region of the population of the second level on the coefficient of exciton-exciton annihilation $\alpha $ and detuning $\overline{\Delta}$.

**Figure 4.**The population of the second excited level as a function of frequency detuning for $\alpha =5$ (bistability) and 20 (monostability) and $\overline{\Omega}=1$. The solid blue lines correspond to the stability, the dashed red line corresponds to the unstable intermediate branch, and the solid red line corresponds to the instability region on the lower branch (left) or in the case of monostability (right).

**Figure 5.**The dynamics of the ground state population in the regime of collision of a pair of counter-propagating switching waves. Arrows show the direction of fronts of switching waves propagation. Time moments t = 0 (the widest distribution), 120 and 440 (stable soliton, the narrowest distribution); $N=300$, $\overline{\Omega}=0.95$.

**Figure 6.**Profiles of the ground state population for stationary solitons for the Rabi frequency $\overline{\Omega}=0.45$ (1), 0.48 (2), 0.78 (3), and 0.95 (4).

**Figure 7.**Profiles of population of the excited state for time moments $t=0$ (

**a**), 50 (

**b**), 300 (

**c**), and 700 (

**d**), see the text.

**Figure 9.**Instantaneous profile of population of excited state in the mode of switching wave (

**a**) and dependence of the wave velocity on the radiation intensity (

**b**).

**Figure 10.**The schemes supporting topological laser solitons. (

**a**) A wide-aperture laser with saturable absorber and mirrors M; x is a transverse Cartesian coordinate. (

**b**) Cavityless scheme. NLM is nonlinear medium with saturable amplification and absorption, pump is incoherent.

**Figure 11.**The upper and lower rows: isointensity surfaces of tangle laser solitons at intensity level $0.5{I}_{\mathrm{max}}$. Two middle rows: Skeletons of the solitons–arrays of their vortex lines. The number of unclosed vortex lines is one (

**a**–

**e**), two (

**j**) and three (

**f**–

**i**). Unclosed lines (

**a**) are absent, “precesson”, (

**b**)–(

**e**)—one unknotted closed, “apples”, (

**f**)—two knotless unlinked, (

**g**)—one, trivial knot, (

**h**)—two unknotted, a single Hopf link, (

**i**)—one knotted, a trefoil knot, (

**j**)—two unknotted, with a double link, “Solomon link”. The torsion index of the closed lines $s=0$ (

**b**–

**f**), —1 (

**g**), —2 (

**h**), —3 (

**i**), —4 (

**j**). The arrows on the vortex lines indicate the direction of the increasing phase of the radiation ($m=1$ ).

**Figure 12.**Domains of electromagnetic energy sources (red) and sinks (blue) for solitons shown in Figure 11 with the same labels.

**Figure 13.**Energy flows for the “precesson”. The only vortex line 1 is oriented according to red arrows ($m=1$). It includes three special points where tangential component of energy flow changes sign. Around the special points there are closed lines of energy flow 2, 3, 4 (2 and 3 are unstable and 4 is saddle limit cycles). Vortex tube is a boundary surface of domain of attraction of energy flow lines in neighborhood of vortex lines 1. It is formed by separatrix energy flow lines beginning on unstable limit cycles 2 or 3 and ending on stable limit cycle 5 (like trajectories 6 and 7) or going to the periphery (like 8 and 9).

**Figure 14.**Domains of stability of three types of tangle laser solitons: “apple”, “Hopf+”, and “trefoil+”. The domains overlap; those for “Hopf+” and “trefoil+” are inside the widest domain of “apple” solitons.

**Figure 15.**Hysteretic dependence of difference of two main inertia moments of “apple” laser solitons with the same topology. In state 1 soliton has fixed “solid-like” structure, whereas it is more asymmetric and oscillates in state 2.

**Figure 16.**Two types of topological reactions with vortex lines: their reconnection (

**a**–

**d**) and separation of loops from the parent line (

**e**–

**i**). Evolution coordinate z increases from (

**a**) to (

**d**) and from (

**e**) to (

**i**).

**Figure 17.**(

**a**) Cycle of variation of small-signal gain ${g}_{0}$ (red trapezoidal line). Inserts: isointensity surfaces and skeletons for “Hopf+” (left, increase of ${g}_{0}$) and “apple” (right, decrease of ${g}_{0}$) solitons. (

**b**) Variation of the field energy ${W}_{f}$ (left) and energy of the medium $d{W}_{m}$ (right) during increase (red) and decrease (green) of ${g}_{0}$.

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**MDPI and ACS Style**

Rosanov, N.N.; Fedorov, S.V.; Nesterov, L.A.; Veretenov, N.A. Extreme and Topological Dissipative Solitons with Structured Matter and Structured Light. *Nanomaterials* **2019**, *9*, 826.
https://doi.org/10.3390/nano9060826

**AMA Style**

Rosanov NN, Fedorov SV, Nesterov LA, Veretenov NA. Extreme and Topological Dissipative Solitons with Structured Matter and Structured Light. *Nanomaterials*. 2019; 9(6):826.
https://doi.org/10.3390/nano9060826

**Chicago/Turabian Style**

Rosanov, Nikolay N., Sergey V. Fedorov, Leonid A. Nesterov, and Nikolay A. Veretenov. 2019. "Extreme and Topological Dissipative Solitons with Structured Matter and Structured Light" *Nanomaterials* 9, no. 6: 826.
https://doi.org/10.3390/nano9060826