Dark spatial soliton and quasi-soliton by arbitrary initial beam profiles in negative Kerr local and nonlocal medium

Journal Pre-proof Dark spatial soliton and quasi-soliton by arbitrary initial beam profiles in negative Kerr local and nonlocal medium ´ Majid Hesami, Mahrokh Avazpour, Mendez Otero, M.D. Iturbe Castillo

PII:

S0030-4026(19)31790-5

DOI:

https://doi.org/10.1016/j.ijleo.2019.163892

Reference:

IJLEO 163892

To appear in:

Optik

Received Date:

20 November 2019

Revised Date:

24 November 2019

Accepted Date:

24 November 2019

Please cite this article as: Hesami M, Avazpour M, Otero M, Iturbe Castillo MD, Dark spatial soliton and quasi-soliton by arbitrary initial beam profiles in negative Kerr local and nonlocal medium, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163892

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.

Dark spatial soliton and quasi-soliton by arbitrary initial beam profiles in negative Kerr local and nonlocal medium Majid Hesami1*, Mahrokh Avazpour2, M.M. Méndez Otero2, M. D. Iturbe Castillo3

1-Instituto de Física”Ing. Luis Rivera Terrazas” Av. San Claudia Y BLVD. 18 sur, Col. San Manuel, C.P. 72570, Puebla, Puebla, México 2- Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur. Col San Manuel, C.P. 72570, Puebla, Puebla, México 3- Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro # 1, 72840 Tonantzintla, Puebla, México

[email protected]

of

*

ur na

lP

re

-p

ro

Abstract: In this paper numerically, generation of dark spatial solitons and quasi-solitons in negative Kerr medium which is described by the (1+1)-Dimensional local and nonlocal Nonlinear Schrodinger Equation, using arbitrary initial condition field distributions dissimilar to the analytical solution is demonstrated. In the local and nonlocal case, different initial beam profiles with different initial beam-widths are considered, while different response functions of medium and degrees of nonlocality are governing the nonlocality. For the local case extra (lack of) energy in the center of beam profile is released to (absorbed from) the symmetric transversal area of media respectively, until the beam profile self evolves to the fundamental dark soliton with its appropriate amplitude and beam-width of the analytical dark soliton hyperbolic tangent. In the nonlocal case, the beam profile finds some oscillation on beam-width and evolves to a Reversed-Gaussian function, regardless of the nonlocal response function and the degree of nonlocality. By increasing the degree of nonlocality, the evolved beam profile finds the wider beam-width. Analytical formulas for the propagated intensity profile in nonlocal media are given. All the numerical simulations are done by MATLAB program and Split-Step method. Keyword: dark soliton, hyperbolic tangent, nonlocality, degree of nonlocality, spatial soliton

Jo

Introduction Solitary optical beams, temporal and spatial solitons, have been the subjects of intense theoretical and experimental studies in recent years. Soliton pulses and their spectra are used for different applications such as optical communication and switching [1][2][3]. Localized and bounded selfguided beams in space-spatial solitons- evolve from an induced nonlinear change of the refractive index in material obtained by the high light intensity distribution. When the effects of the induced refractive nonlinearity and the beam diffraction exactly compensate each other, the beam propagates without change in shape and is said to be self-trapped [4][5]. The change of induced local refractive index is directly proportional to the light intensity. Paraxial evolution of the beam is governed by famous cubic nonlinear Schrodinger (NLS) equation. Depending on the sign of Kerr coefficient the nonlinearity may be either positive (negative), for self-focusing (selfdefocusing) nonlinear medium with solution of spatial bright (dark) soliton respectively. In this paper the self-defocusing nonlinearity is considered, which corresponds to a negative Kerr 1

+∞

𝑅(𝑋 ′ − 𝑋)𝐼(𝑋 ′ , 𝑍)𝑑𝑋′

(1)

re

Δ𝑛(I) = − ∫

-p

ro

of

coefficient. The well-known fundamental dark soliton is obtained by propagation of the initial beam profile of hyperbolic tangent in negative Kerr medium [6][7]. In local medium, the change of induced refractive index at a particular point is proportional to the intensity in that exact point, however in nonlocal medium, light-induced refractive index change is determined by the light intensity in a certain neighborhood of this location [8]. The nonlocality is common to many nonlinear systems, e.g. in nematic liquid crystals [9], media with thermal nonlinearity [10], atomic vapors [11], plasmas [12], Bose-Einstein condensates [13], etc. Specially in optics, nonlocality occurs due to different physical mechanism [14], such as long-range forces [15], transport [16], or many body interaction [17][18]. The interest over the dynamic behaviour of nonlocal solitons has been grown significantly and receives well attention by researchers [19][20][21] [22][23] to study new problems, by variety of methods [24][25][26][27][28]. The most accepted mathematical model to describe the effect of nonlocality is to consider function 𝑅(𝑋) where characterizes the nonlocal response of the medium and it is assumed to be real and symmetric. The most important effect of nonlocality is on the change of refractive index, Δ𝑛, in the (1+1)-Dimensional Nonlinear Schrodinger (NLS) equation. This effect on Δ𝑛 is described in Eq.(1) as a general phenomenological model for self-defocusing Kerr-like media, by the convolution integral between nonlocal response function of medium 𝑅 (𝑋) and the local intensity 𝐼(𝑋) of the beam [29][30][31][32].

−∞

Jo

ur na

lP

The width of the response function is related to the degree of nonlocality, from very small value as Dirac delta indicating the local medium till high value implying the highly nonlocality [29]. Hyperbolic tangent (Tanh) as an initial beam profile is the recognized solution of NLS equation in local negative Kerr medium, where evolves as a fundamental dark soliton [7], however for nonlocal medium when different degrees of nonlocality are considered the exact analytical solutions [29][33][34][35][36][37][38] can be very complicated. There are methods ([38][39] and references there in) to obtain the appropriate initial dark soliton profile, however lower attention has been given to evaluate how an arbitrary initial beam profile evolves by propagation in one dimensional negative nonlinear medium with different degrees of nonlocality. In this paper we are going to numerically simulate the propagation of some symmetric arbitrary initial beam profiles different of hyperbolic tangent (Tanh) in local and nonlocal negative nonlinear medium. At the first the evolution of two initial profiles of Reversed Rectangular Function (RRF) and Reversed Triangular Function (RTF) with two different initial beam-widths (very small, and very wide) in local medium is considered. And it is demonstrated that, regardless of the initial beam profiles and beam-widths, the intensity profile evolves to a fundamental Tanh dark soliton by propagation. In addition, in nonlocal medium with arbitrary nonlocal response functions and with different degrees of nonlocality, the intensity profile of arbitrary initial beam profile evolves to Reversed Gaussian function, and its analytical formula is presented.

Numerical simulation in Local negative Kerr medium The propagation process in linear medium is developed to the induced third order nonlinear medium, when the intensity of initial beam profile is sufficiently high. In third order nonlinear 2

(Kerr) medium the refractive index of medium is not a constant value and depends on the intensity of the beam in the form of 𝑛(𝐼 ) = 𝑛0 + 𝑛2 𝐼, where 𝑛0 the linear refractive index and 𝑛2 is the Kerr-coefficient [40]. The paraxial evolution of one dimensional transversal beam profile in the form of 𝐸 = 𝐸 (𝑋, 𝑍) 𝑒𝑥𝑝(𝑖𝐾𝑍 ) propagating in the induced third order nonlinear medium in Zdirection is governed by (1+1)-Dimensional Nonlinear Schrodinger (NLS) equation, which after some normalization we write it in the form of the following: 𝜕𝐴(𝑋, 𝑍) 1 𝜕 2 𝐴(𝑋, 𝑍) 𝐿𝐷 (2) −𝑖 = + |𝐴(𝑋, 𝑍)|2 𝐴(𝑋, 𝑍) 𝜕𝑍 4 𝜕𝑋 2 𝐿𝑁𝐿 where A(𝑋, 𝑍) is a complex envelop function normalized to the initial maximum intensity√𝐼𝑚 , 𝑧 that measures the complex field amplitude with 𝑋 = 𝑥/𝑥0 and 𝑍 = 𝐿 being the normalized

of

𝐷

𝐿𝑁𝐿

ro

transverse and longitudinal coordinates respectively. 𝑥0 is the initial beam-width, 𝐿𝐷 = (𝑛0 𝑘0 𝑥02 )/2 is the Rayleigh distance or diffraction length, with 𝑘0 = 2𝜋/𝜆 the wave vector, 𝜆 the wavelength, 𝐿𝑁𝐿 = (𝑛2 𝑘0 𝐼𝑚 )−1 the self-focusing distance. Due to sign of the Kerr-coefficient 𝐿 𝐿 n2 > 0 (𝑛2 < 0) being positive (negative), the relation 𝐷 = 1 ( 𝐷 = −1), and Eq. (2) allows 𝐿𝑁𝐿

lP

re

-p

analytical stationary fundamental solutions known as bright (dark) spatial solitons described by hyperbolic secant (hyperbolic tangent) respectively. However, in this paper just negative Kerrcoefficient (𝑛2 < 0) in NLS equation is considered and consequently the third order negative Kerr medium is under investigation. So that the evolution of one-dimensional Tanh function with the specific normalized amplitude and initial beam-width Eq. (3) in negative Kerr medium can propagates as fundamental dark spatial soliton, other wise by different initial beam-width, the beam can propagates other order of dark soliton [41]. 𝑋 (3) ) 0.7071 The evolution of one-dimensional Tanh beam profile (see Eq. (3)) is shown in Fig. 1. The figure shows the propagation in Z-direction for 10 Rayleigh distance (10Z), which the beam-width, the amplitude profile and the intensity profile are not suffering any changes. Obtaining the constant intensity profile by propagation in negative Kerr medium is called dark soliton. The dark soliton occurs due to influence of the self-defocusing effect, which balancing the spatial diffraction. Fig. 1(a) shows the top view of the dark soliton propagation where the beam launches in the medium from the down-side, and the color is referred to the intensity value, while Fig. 1(b) demonstrates the same information, just the intensity is shown in perpendicular axis.

Jo

ur na

𝐴(𝑋) = 1 ∗ 𝑇𝑎𝑛ℎ (√2 𝑋) = 1 ∗ 𝑇𝑎𝑛ℎ (

a)

b)

Fig. 1:Evolution of one-dimensional Tanh beam through the negative Kerr medium

3

However, in this paper the propagation of one-dimensional arbitrary initial beam profiles different than Tanh in the negative Kerr medium is investigated. The Eqs. (4) and (5) which we call them, Reversed Rectangular Function (RRF) and Reversed Triangular Function (RTF) are considered as two arbitrary initial beam profiles. In these equations 𝐴𝑅 (𝑋) and 𝐴 𝑇 (𝑋) are the normalized amplitude of the initial beam profile, with the initial beam-width of 𝑏𝑅 and 𝑏𝑇 . The beam profiles in Eqs. (4) and (5) are multiplied to a tanh(𝑋/0.001), for the aim of giving the 𝜋 retardant phase to the half transversal side (see Fig. 2 row (a)).The Split-Step method is used to numerically simulate the propagation by the MATLAB program. 𝑋 ) ) ∗ tanh(𝑋/0.001) 𝑏𝑅

(4)

of

𝐴𝑅 (𝑋) = (1 − 𝑟𝑒𝑐𝑡𝑝𝑢𝑙𝑠 (

ro

The Split-Step method also know as Split-Step Fourier method was introduced by Fisher [42], helps to calculates the propagation of an optical beam in the nonlinear and dispersive medium, by dividing the medium along the propagation direction in many small sections. All the sections are split in two parts, in first part the nonlinearity property and in the second part the diffraction property is absent [43][44]. 𝑋 ) ) ∗ tanh(𝑋/0.001) (5) 𝑏𝑇 In order to demonstrate that the paraxial evolution of any symmetrical initial beam profile in negative Kerr medium evolves to a |tanh|2 intensity profile, first, a very wide initial beam widths of 𝑏𝑅 = 2 and 𝑏𝑇 = 2 for both profiles of RRF and RTF in Eqs. (4) and (5) are considered. Then, the very small value of initial beam widths (𝑏𝑅 = 0.2 & 𝑏𝑇 = 0.2) for both profiles are studied.

lP

re

-p

𝐴 𝑇 (𝑋) = (1 − 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟𝑃𝑢𝑙𝑠𝑒 (

Jo

ur na

In Fig. 2, and Fig. 3 the evolution of each distribution along Z-direction for 20 Rayleigh distance, 20 Z, together with its initial amplitude, initial and final intensity profile is demonstrated. In Fig. 2 the higher initial beam-width is chosen (𝑏𝑅 = 2 & 𝑏𝑇 = 2), whereas in Fig. 3 the lower initial beam-width is selected (𝑏𝑅 = 0.2 & 𝑏𝑇 = 0.2). As initial beam profile in both figures, in the left column, the RRF, and in the right column the RTF is used. In Fig. 2 row (a) the initial amplitudes of both profiles are plotted. Fig. 2 (row b & C) and Fig. 3 (row a & b) are demonstrating the twoand three-dimensional view of intensity propagation, where in two-dimensional view, the beam launched from down side. Fig. 2 row (d) and Fig. 3 row (c) plot the comparison between initial (blue dashed line) and final (red solid line) intensity profiles as well as Tanh-Test (green marks) intensity profile. For plotting the Tanh-Test intensity profile the fundamental dark soliton beam profile mentioned in Eq. (3) is used. Fig. 2 and Fig. 3 demonstrate that, the two arbitrary considered initial beam profiles (RRF,RTF) evolve to the fundamental dark soliton by propagation in negative Kerr medium. In addition, the figures show that, the initial beam-width doesn’t have much effect over producing the dark solitons. When the initial beam-width is very small 𝑏𝑅 = 0.2 & 𝑏𝑇 = 0.2 (very wide 𝑏𝑅 = 2 & 𝑏𝑇 = 2), initially, high (low) intensity value is located in the center of transversal axis, and by initiating the propagation process, the medium, takes this extra intensity away from the transversal center (substitute the lack of energy from surrounding sides to the center), until the suitable width for Tanh is obtained respectively. In other words, by using the arbitrary symmetric initial beam profile, holding arbitrary beam-width, after some initial step of propagation, the intensity profile reshapes to a dark soliton with suitable width in the form of Tanh profile mentioned in Eq. (3). When a wide initial beam-width is considered, the beam takes some 4

distance of propagation until reshapes itself to Tanh intensity profile at the center of the beam (see Fig. 2 row (b & c)), however for the case of narrow initial beam-width, reshaping the intensity profile occurs in a very short propagation distance (see Fig. 3 row (a & b)). So that, in negative local Kerr medium the value of initial beam-width is not a big deal for producing the dark soliton, since the medium self-adjust its beam-width to Tanh.

ro

of

a)

re

-p

b)

d)

ur na

lP

c)

Jo

Fig. 2: (left column) Reversed Rectangular Function for 𝑏𝑅 = 2 , and (right column) Reversed Triangular Function for 𝑏𝑇 = 2 as the initial field distribution. (a) initial amplitude, (b) 2-D, (c) 3-D intensity propagation, and (d) intensity profiles in linear: initial (blue dashed line) and final (red-solid line) distributions. Tanh-Test (Green marks).

5

a)

-p

ro

of

b)

lP

re

c)

ur na

Fig. 3: (left column) RRF for 𝑏𝑅 = 0.2 , and (right column) RTF for 𝑏𝑇 = 0.2 as the initial field distribution. (a) 2-D, (b) 3-D intensity propagation, and (c) intensity profiles in linear: initial (blue dashed line) and final (red-solid line) distributions. Tanh-Test (Green marks).

Numerical simulation in nonlocal negative Kerr medium For numerical simulation of propagation in nonlocal media, the RTF and RRF as arbitrary initial beam profile mentioned in Eqs. (4) and (5) are used. The following nonlocal responses of medium Eqs. (6) and (7) are considered, where 𝑅𝑠 (𝑋) and 𝑅𝑔 (𝑋) are the hyperbolic secant (Sech) and gaussian nonlocal response functions. 𝑅0𝑠 and 𝑅0𝑔 are normalization factors obtained by +∞

Jo

∫−∞ 𝑅(𝑋)𝑑𝑋 = 1 , and 𝛼𝑠 and 𝛼𝑔 are the degree of nonlocality for Sech and gaussian nonlocal responses respectively. 𝑅𝑠 (𝑋) = 𝑅0𝑠 𝑆𝑒𝑐ℎ (

+∞ 𝑋 𝑋 ) → 𝑅0𝑠 = 1/ ∫ 𝑆𝑒𝑐ℎ ( ) 𝑑𝑋 𝛼𝑠 𝛼𝑠 −∞

(6)

2

𝑋 1 𝑅𝑔 (𝑋) = R 0g exp (− ( ) ) → R 0g = 𝛼𝑔 √𝜋𝛼𝑔

(7)

6

In Fig. 4, and Fig. 5 initial beam profile of RTF and RRF with very narrow initial beam-width of 𝑏𝑇 = 0.2 and 𝑏𝑅 = 0.2 are used respectively. In both figures in row (a) , (b) and (c) the degree of nonlocality 𝛼𝑠 =1, 5 and 10 is adjusted correspondingly. In addition, in left column 2D and middle column 3D simulation of propagation is demonstrated for 100Z distance of propagation. In right column of both figures the initial and final intensity profile, as well as the intensity of our proposed Reversed-Gaussian-test is plotted. The intensity profile of the Reversed-Gaussian-test, obtained by using the two first absolute maximum values of propagated amplitude profile around the transversal center, |𝐴𝑚 (𝑍 )|, at 𝑋𝑖 and 𝑋𝑓 . The power of propagated profile, 𝑃𝑓 = 𝑋

∫𝑋 𝑓 |𝐴(𝑋, 𝑍)|2 𝑑𝑋 , in these limited transversal points are calculated. To test either the final 𝑖

of

intensity profile is a Reversed-Gaussian-test profile or not the following intensity profile is proposed. 2

ro

𝑋 2 (8) 𝐼𝑔 (𝑋) = [|𝐴𝑚 (𝑍)| (1 − exp (− ( ) ))] 𝑏𝑋 Where the 𝑏𝑋 =? is unknown. The final power , 𝑃𝑓 , and the power of Reversed-Gaussian-test is compared in following,

𝑋𝑖

𝑋𝑖

2

-p

∫ 𝐼𝑔 (𝑋)𝑑𝑋 = ∫

𝑋𝑓

𝑋 2 [|𝐴𝑚 (𝑍)| (1 − exp (− ( ) ))] 𝑑𝑋 = 𝑃𝑓 𝑏𝑋 2(𝑋𝑓 − 𝑋𝑖 ) − |

2𝑃𝑓 𝐴𝑚 (𝑍)|2

√𝜋(4 − √2)

lP

𝑏𝑋 =

re

𝑋𝑓

Jo

ur na

The right column of Fig. 4 and Fig. 5 show that, the final intensity profile has a good match with intensity profile of the Reversed-Gaussian-test.

7

a)

-p

ro

of

b)

lP

re

c)

Jo

ur na

Fig. 4: Numerical propagation for 100Z, in 2D (left column), and 3D (middle column) and intensity profiles in linear (right column) for a RTF as initial field, with initial beam-width 𝑏𝑇 = 0.2 in a media with Sech nonlocal response with degrees of: (a) 𝛼𝑠 = 1, (b) 𝛼𝑠 = 5, and (c) 𝛼𝑠 = 10. Input intensity profile (blue dashed line), Final intensity profile (red) and Reversed-Gaussian-test (green marks).

8

a)

ro

of

b)

re

-p

c)

lP

Fig. 5: Numerical propagation for 100Z in 2D (left column), and 3D (middle column) and intensity profiles in linear (right column) for a RRF as initial field, with initial beam-width 𝑏𝑅 = 0.2 in a media with Sech nonlocal response with degrees of: (a) 𝛼𝑠 = 1, (b) 𝛼𝑠 = 5, and (c) 𝛼𝑠 = 10. Input intensity profile (blue dashed line), Final intensity profile (red) and Gaussian test (green marks).

Jo

ur na

To see how the beam profile evolves with wider initial beam-width, the initial beam profiles of RTF and RRF with 𝑏𝑇 = 2 and 𝑏𝑅 = 2 in Sech nonlocal response function is considered. The result of numerical propagation is demonstrated in Fig. 6. In left column of this figure, RTF as initial beam profile and in the right column RRF as initial beam profile is used. Input (blue dashed line), final (red solid line) and Reversed-Gaussian-test (green marks) intensity profiles are plotted, where in Fig. 6 row (a , b & c) the nonlocal degrees of 𝛼𝑠 =1, 5 and 10 are considered individually. The result shows that there is a good match between final intensity profile and proposed ReversedGaussian-test intensity profile.

9

a)

-p

ro

of

b)

lP

re

c)

ur na

Fig. 6: RTF (left column), RRF (right column) with same initial beam-width of 𝑏𝑇 = 2 and 𝑏𝑅 = 2 as initial conditions are used, in a media with Sech nonlocal response with degrees of: (a) 𝛼𝑠 = 1, (b) 𝛼𝑠 = 5, and (c) 𝛼𝑠 = 10. Intensity profiles in linear for input (blue dashed line), final (red) and Reversed-Gaussian-test (green marks) intensity profile after 100 Z propagation distance.

Jo

In Fig. 7 the Gaussian nonlocal response function as mentioned in Eq. (7) is employed with different degrees of nonlocality, where the propagation of RTF as initial field with wide initial beam-width 𝑏𝑇 = 2 is demonstrated. In Fig. 7 row (a, b & c) the nonlocal degrees of 𝛼𝑔 =1, 5 and 10 respectively are considered. It can be clearly seen, there is not too much differences when different nonlocal response functions are considered, and just the radius effect of nonlocal response function is important. In addition, when considering equal value on degree of nonlocality for different nonlocal response functions, it doesn’t mean the same effect of degrees are chosen, for example same value of degree of nonlocality for gaussian and Sech ( 𝛼𝑠 =5, 𝛼𝑔 =5 ) has not same degree of nonlocality effect [45]. The Fig. 7 shows that, the intensity profile in a medium with gaussian nonlocal response function also, reshapes to proposed Reversed-Gaussian-test intensity profile as demonstrated in left column of figure.

10

a)

ro

of

b)

re

-p

c)

lP

Fig. 7: Numerical propagation for 100Z, in 2D (left column), and 3D (middle column) and intensity profiles in linear (right column) for a RTF as initial field, with initial beam-width 𝑏𝑇 = 2 in a media with Gaussian nonlocal response and nonlocal degrees 𝛼𝑔 of: (a) 𝛼𝑔 = 1, (b) 𝛼𝑔 = 5, and (c) 𝛼𝑔 = 10. Input intensity profile (blue dashed line), Final intensity profile (red) and Reversed-Gaussian-test (green marks).

Jo

ur na

The results of propagation in low degree of nonlocality show, for narrow (wide) initial beamwidth, the beam profile during very initial steps of the propagation start to self-expand (selfreduce) its beam-width by releasing (absorbing) energy in symmetric transversal sides respectively, until reaches the almost constant intensity profile shape ,Reversed-Gaussian-Test, (see row (a) Fig. 4, Fig. 5, and Fig. 7) regardless of initial beam-width or profile. In contrast, by increasing the degree of nonlocality with respect to intensity oscillation, the exchange of energy with transversal sides grows. In addition, the beam-width is increased as the degree of nonlocality increases. For the same degrees of nonlocality, with the equal distance of propagation; the beam profile of Reversed-Gaussian-Test, gains the same beam-width and amplitude, regardless of initial condition of beam profile. The Table 1 gives data about the profile of propagated beam in the form of the Reversed-Gaussian-Test profile after 100 Rayleigh distance in Sech nonlocal response medium. The information in this table is obtained from the profiles in green marks in Fig. 4, Fig. 5 and Fig. 6. The 𝑏𝑥𝑇 and |𝐴𝑚 | 𝑇 (𝑏𝑥𝑅 and |𝐴𝑚 |𝑅 ) are the width and amplitude of the ReversedGaussian-Test when RTF (RRF) as initial condition is used respectively. From the table it can be seen that, when the same degree of nonlocality is chosen, the parameters of the Reversed-GaussianTest (amplitude and width) is same for any case of initial condition.

11

𝑏𝑅 = 𝑏𝑇 = 0.2

𝑏𝑅 = 𝑏𝑇 = 2

𝑏𝑥𝑇 𝛼𝑠 = 1 0.7065 𝛼𝑠 = 5 1.6500 𝛼𝑠 = 10 2.6474

|𝐴𝑚 | 𝑇 1.0962 1.2131 1.2076

0.7065 1.6501 2.6476

|𝐴𝑚 |𝑅 1.0962 1.2131 1.2076

𝛼𝑠 = 1 0.7065 𝛼𝑠 = 5 1.7269 𝛼𝑠 = 10 2.8665

1.0960 1.2169 1.1942

0.7064 1.7152 2.8158

1.0959 1.2149 1.1952

𝑏𝑥𝑅

Table 1: beam-width and amplitude of propagated beam filed in Sech nonlocal medium for 100 Rayleigh distance

ur na

lP

re

-p

ro

of

Conclusion In this paper we have studied the numerical propagation of some arbitrary initial field distributions in negative local and nonlocal nonlinear media described by the (1+1)-Dimensional Nonlinear Schrodinger Equation. In the local part, we demonstrated that any symmetric and arbitrary initial beam profiles different from hyperbolic tangent distribution, can propagates as a dark soliton. As evidence two initial profiles were presented: Reversed Triangular Function and Reversed Rectangular Function with very small and very wide initial beam-widths. The results of simulation for local case show, when very narrow initial beam-width is chosen, by the initiating the propagation, the beam immediately reshapes to the dark soliton, however by using wide initial beam-with, it takes some distance of propagation until evolves to the dark soliton. In addition, it is demonstrated that, this reshaping to dark soliton has no relationship to initial beam profiles, nor to the initial beam-widths, just for small (wide) initial beam-width, some portion of energy is released to (absorbed from) transversal areas symmetrically. As the result, whole energy of arbitrary initial beam profile is not confined in direction of propagation. The propagated intensity profile is compared with Tanh intensity profile. In the nonlocal part, same initial beam profiles and beam-widths have been considered, and the nonlocality is applied by different response functions of medium and degrees of nonlocality. In this nonlocal medium, the beam profile evolves to our proposed Reversed-Gaussian-test function, regardless of the nonlocal response function, degrees of nonlocality, initial beam profile, and the initial beam-widths. The results in nonlocal medium show, by increasing the degree of nonlocality, the evolved beam profile finds the wider beamwidth. Analytical formula for our proposed final intensity profile shape is given, and it is compared with the final intensity profile, where shows a very good fit.

Jo

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement Thanks, Consejo Nacional de Ciencia y Tecnología (CONACYT) Grant 716213.

12

References Fatome J, Kibler B, Andresen E R, Rigneault H and Finot C 2012 All-fiber spectral compression of picosecond pulses at telecommunication wavelength enhanced by amplitude shaping Appl. Opt. 51 4547–53

[2]

Avazpour M, Beltrán-Pérez G, Rodríguez-Morales L A, Armas-Rivera I, Ibarra-Escamilla B, Muñoz-Aguirre S, Castillo-Mixcóatl J, Pottiez O and Kuzin E A 2019 The use of polarization-imbalanced NOLM to improve the quality of the spectrum compression Opt. Laser Technol. 120 105692

[3]

Kivshar Y S and Stegeman G I 2002 Spatial optical solitons Opt. Photonics News 13 59– 63

[4]

Agrawal G P 2000 Nonlinear fiber optics Nonlinear Science at the Dawn of the 21st Century (Springer) pp 195–211

[5]

Agrawal G P 2012 Fiber-optic communication systems vol 222 (John Wiley & Sons)

[6]

Zhao 5 _W and Bourkoff E 1989 Propagation properties of dark solitons Opt. Lett. 14 703– 5

[7]

Kivshar Y S and Luther-Davies B 1998 Dark optical solitons: physics and applications Phys. Rep. 298 81–197

[8]

Królikowski W, Bang O, Nikolov N I, Neshev D, Wyller J, Rasmussen J J and Edmundson D 2004 Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media J. Opt. B Quantum Semiclassical Opt. 6

[9]

Peccianti M, Conti C, Assanto G, De Luca A and Umeton C 2004 Routing of anisotropic spatial solitons and modulational instability in liquid crystals Nature 432 733

[10]

Rotschild C, Alfassi B, Cohen O and Segev M 2006 Long-range interactions between optical solitons Nat. Phys. 2 769

[11]

Skupin S, Saffman M and Krolikowski W 2007 Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor Phys. Rev. Lett. 98 263902

[12]

Litvak A G, Mironov V A, Fraiman G M and Iunakovskii A D 1975 Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity Fiz. Plazmy 1 60–71

[13]

Pedri P and Santos L 2005 Two-dimensional bright solitons in dipolar Bose-Einstein condensates Phys. Rev. Lett. 95 200404

Jo

ur na

lP

re

-p

ro

of

[1]

[14]

Kaminer I, Rotschild C, Manela O and Segev M 2007 Periodic solitons in nonlocal nonlinear media Opt. Lett. 32 3209

[15]

Conti C, Peccianti M and Assanto G 2004 Observation of Optical Spatial Solitons in a Highly Nonlocal Medium Phys. Rev. Lett. 92 113902–1

[16]

Ultanir E A, Stegeman G I, Lange C H and Lederer F 2004 Coherent interactions of dissipative spatial solitons Opt. Lett. 29 283–5

[17]

Dalfovo F, Giorgini S, Pitaevskii L P and Stringari S 1999 Theory of Bose-Einstein 13

condensation in trapped gases Rev. Mod. Phys. 71 463 Pecseli H L and Rasmussen J J 1980 Nonlinear electron waves in strongly magnetized plasmas Plasma Phys. 22 421

[19]

Biswas A, Y\ild\ir\im Y, Ya\csar E, Zhou Q, Alshomrani A S and Belic M 2019 Optical soliton perturbation in parabolic law medium having weak non-local nonlinearity by a couple of strategic integration architectures Results Phys. 13 102334

[20]

Horikis T P and Frantzeskakis D J 2016 Asymptotic reductions and solitons of nonlocal nonlinear Schrödinger equations J. Phys. A Math. Theor. 49 205202

[21]

Wang Q, Li J, Gao X and Xie W 2018 Nonlocal logarithmic nonlinear optical soliton Optik (Stuttg). 172 571–7, https://doi.org/10.1016/j.ijleo.2018.06.089

[22]

Jawad A J M, Abu-AlShaeer M J, Majid F B, Biswas A, Zhou Q and Belic M 2018 Optical soliton perturbation with exotic non-Kerr law nonlinearities Optik (Stuttg). 158 1370–9, https://doi.org/10.1016/j.ijleo.2017.12.184

[23]

Zhang H, Chen F, Yu C and Sun L 2018 Dynamic behaviors of nonlocal solitons based on initial spatial chirp Optik (Stuttg). 174 372–81, https://doi.org/10.1016/j.ijleo.2018.08.075

[24]

Biswas A, Yildirim Y, Yasar E, Triki H, Zhou Q, Moshokoa S P, Ullah M Z, Belic M and others 2018 Optical soliton perturbation with full nonlinearity in polarization preserving fibers using trial equation method J. Optoelectron. Adv. Mater. 20 385–402

[25]

Biswas A, Ekici M, Sonmezoglu A, Mirzazadeh M, Zhou Q, Alshomrani A S, Moshokoa S P and Belic M 2018 Optical solitons in parabolic law medium with weak non-local nonlinearity by extended trial function method Optik (Stuttg). 163 56–61

[26]

Biswas A, Ekici M, Sonmezoglu A and Alqahtani R T 2018 Optical solitons with differential group delay and weak non-local nonlinearity by extended trial function method Optik (Stuttg). 166 31–8

[27]

Biswas A, Rezazadeh H, Mirzazadeh M, Eslami M, Zhou Q, Moshokoa S P and Belic M 2018 Optical solitons having weak non-local nonlinearity by two integration schemes Optik (Stuttg). 164 380–4

[28]

Hubert M B, Justin M, Betchewe G, Doka S Y, Biswas A, Zhou Q, Alshomrani A S, Ekici M, Moshokoa S P and Belic M 2018 Optical solitons in parabolic law medium with weak non-local nonlinearity using modified extended direct algebraic method Optik (Stuttg). 161 180–6

Jo

ur na

lP

re

-p

ro

of

[18]

[29]

Królikowski W and Bang O 2001 Solitons in nonlocal nonlinear media: Exact solutions Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 63 1–6

[30]

Kong Q, Shen M, Chen Z, Wang Q, Lee R K and Krolikowski W 2013 Dark solitons in nonlocal media with competing nonlinearities Phys. Rev. A - At. Mol. Opt. Phys. 87 1–5

[31]

Ge L, Wang Q, Shen M, Shi J, Kong Q and Hou P 2009 Dark solitons in nonlocal media: variational analysis J. Opt. A Pure Appl. Opt. 11 65207

[32]

Nikolov N I, Neshev D, Królikowski W, Bang O, Rasmussen J J and Christiansen P L 2004 14

Attraction of nonlocal dark optical solitons Opt. Lett. 29 286 Mitchell D J and Snyder A W 1999 Soliton dynamics in a nonlocal medium JOSA B 16 236–9

[34]

Bang O, Krolikowski W, Wyller J and Rasmussen J J 2002 Collapse arrest and soliton stabilization in nonlocal nonlinear media Phys. Rev. E 66 46619

[35]

Conti C, Peccianti M and Assanto G 2003 Route to Nonlocality and Observation of Accessible Solitons Phys. Rev. Lett. 91 073901

[36]

Buccoliero D, Desyatnikov A S, Krolikowski W and Kivshar Y S 2007 Laguerre and Hermite soliton clusters in nonlocal nonlinear media Phys. Rev. Lett. 98 53901

[37]

Kong Q, Wang Q, Bang O and Krolikowski W 2010 Analytical theory for the dark-soliton interaction in nonlocal nonlinear materials with an arbitrary degree of nonlocality Phys. Rev. A - At. Mol. Opt. Phys. 82 1–6

[38]

Duque E I and Lopez-Aguayo S 2019 Generation of solitons in media with arbitrary degree of nonlocality using an optimization procedure Phys. Rev. A 99 13831

[39]

Ablowitz M J and Musslimani Z H 2005 Spectral renormalization method for computing self-localized solutions to nonlinear systems Opt. Lett. 30 2140–2

[40]

Kivshar Y S and Agrawal G 2003 Optical solitons: from fibers to photonic crystals (Academic press)

[41]

Hesami M, Avazpour M and Otero M M M 2019 Transforming higher order bright and dark solitons to the first order solitons in Kerr medium: A review Optik (Stuttg). 163695, https://doi.org/10.1016/j.ijleo.2019.163695

[42]

Fisher R A and Bischel W K 1975 Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses J. Appl. Phys. 46 4921–34

[43]

López-Aguayo S, Ochoa-Ricoux J P and Gutiérrez-Vega J C 2006 Exploring the behavior of solitons on a desktop personal computer Rev. Mex. Fis. E 52 28–36

[44]

Agrawal G 2001 Applications of nonlinear fiber optics (Elsevier)

[45]

Hesami M, Avazpour M, Otero M M M, Arriaga J, Castillo M D I and Cerda S C 2020 Generation of bright spatial quasi-solitons by arbitrary initial beam profiles in local and nonlocal (1+ 1)-Dimensional nonlinear media Optik (Stuttg). 202 163504, https://doi.org/10.1016/j.ijleo.2019.163504

Jo

ur na

lP

re

-p

ro

of

[33]

15