Generation of bright spatial quasi-solitons by arbitrary initial beam profiles in local and nonlocal (1+1)-Dimensional nonlinear media

Journal Pre-proof Generation of Bright spatial quasi-solitons by arbitrary initial beam profiles in local and nonlocal (1+1)-Dimensional nonlinear media ´ Majid Hesami, Mahrokh Avazpour, M.M. Mendez Otero, J. Arriaga, ´ M.D. Iturbe Castillo, S. Chavez Cerda

PII:

S0030-4026(19)31402-0

DOI:

https://doi.org/10.1016/j.ijleo.2019.163504

Reference:

IJLEO 163504

To appear in:

Optik

Received Date:

9 September 2019

Revised Date:

30 September 2019

Accepted Date:

1 October 2019

Please cite this article as: Hesami M, Avazpour M, Otero MMM, Arriaga J, Castillo MDI, Cerda SC, Generation of Bright spatial quasi-solitons by arbitrary initial beam profiles in local and nonlocal (1+1)-Dimensional nonlinear media, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163504

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Generation of Bright spatial quasi-solitons by arbitrary initial beam profiles in local and nonlocal (1+1)-Dimensional nonlinear media Majid Hesami1*, Mahrokh Avazpour2, M.M. Méndez Otero2, J. Arriaga1, M. D. Iturbe Castillo3, S. Chávez Cerda3

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1-Instituto de Física” Ing. Luis Rivera Terrazas”, Benemérita Universidad Autónoma de Puebla, 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.

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*[email protected]

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Abstract: In this paper we demonstrate numerically that it is possible to generate bright spatial quasi-solitons, in media that are described by the (1+1)-Dimensional local and nonlocal Nonlinear Schrodinger Equation, using initial condition field distributions that are distinct from the analytical solution. In the local case different initial conditions were considered and in the nonlocal case different response functions and degrees of nonlocality were considered, obtaining the same behavior: at the beginning the initial profile is reshaped and after some distance, the beam propagates almost without change or in an small oscillatory way. The propagated intensity profile in the local case evolves to hyperbolic secant and in the nonlocal case to Gaussian, regardless of the degree of nonlocality. Analytical formulas for the propagated intensity profile are given.

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Key words: Solitons, local medium, nonlocal medium, quasi-solitons.

1. Introduction

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Self-trapped of optical beams or more precisely Spatial Optical solitons [1] propagate without change through some kind of nonlinear media. They can be generated in media that present an intensity-dependent refractive index [2], where diffraction is compensated by the nonlinear effect. While in local medium limit, the change of refractive index in one point depends on the intensity in that point, in nonlocal medium, the change of refractive index depends on the intensity in vicinity points [3]. Analytical solutions for such solitons have been obtained from the equations that describe the beam evolution in local [4] and nonlocal media [5-12]. In optics, nonlocality can be due to different physical mechanism [13], such as long-range forces [14], transport [15], or many body interaction [16-17]. In many media or physical systems, naturally nonlocality appears, such as Nematic liquid crystal[18], Bose-Einstein condensates [19], thermal media [20], etc. The most accepted mathematical model to describe the nonlocality of a medium is considering that, change of induced refractive index Δn ,in the (1+1)-Dimensional Nonlinear Schrödinger Equation (NLSE), must be written by convolution integral between nonlocal response function of medium R(X) and the intensity I(X) of beam [3]:

𝑏

Δn(I) = ∫ R(𝑋 ′ − 𝑋)I(𝑋 ′ , 𝑍)𝑑𝑋′

(1)

𝑎

2. Local medium

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where the response function of medium R(𝑋) has the property of being real, localized, and symmetric. Response function width is associated to the degree of nonlocality and can be started from narrow value as Dirac delta until some high value, which corresponds to go from a local medium to a highly nonlocal situation, respectively. It is well known that the Hyperbolic Secant (Sech) initial profile is the adequate to obtain spatial optical soliton in a local medium [4], however for nonlocal medium with different degree of nonlocality the exact analytical solutions [3,5-12] can be very complicated. In [21] a numerical method was developed to obtain soliton solutions in media with arbitrary degree of nonlocality for two type of response functions: Gaussian and exponential. Obtaining that in media with Gaussian response function, for the case of high nonlocality, the soliton profile must be Gaussian. However, for very low nonlocality the Sech profile is the adequate profile for soliton. And they reported that, this issue is not happening in the case of media with exponential response function, even in the case of high nonlocality: Gaussian beam profile is not the adequate for having soliton behavior. Although, depending on the degree of nonlocality, exist methods ( [12,22] and references there in) to find the adequate initial soliton profile little attention has been given to analyze what happen when an arbitrary initial beam profile is propagated in one-dimensional nonlinear media with any degree of nonlocality. In this paper we are going to numerically study the propagation of symmetric arbitrary initial field profiles in local and nonlocal nonlinear media. So that, first of all, we are going to demonstrate that, in local medium arbitrary shape symmetric beam profile with the adequate initial beam-width can propagate as a quasi-soliton and the propagated intensity profile takes the form of square hyperbolic secant. In addition, in nonlocal medium with arbitrary nonlocal response function and with any degree of nonlocality a similar behavior can be obtained: any initial beam profile with suitable initial beam-width can give rise to a quasi-soliton in this case the propagated intensity profile takes the form of square Gaussian. Finally, the conclusions are given.

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The paraxial evolution of a (1+1)-Dimensional beam, with field amplitude E = E(x, z)exp(ikz) propagating along the z-direction in a local Kerr medium ( Δn(I) = n2 I ) is described by the Nonlinear Schrodinger Equation NLSE given by: ∂A(𝑋, 𝑍) 1 ∂2 A(𝑋, 𝑍) LD (2) −𝑖 = + |A(𝑋, 𝑍)|2 A(𝑋, 𝑍) ∂Z 4 ∂𝑋 2 LNL 1/2 where A(𝑋, 𝑍) is the field amplitude normalized to the maximum intensity Im , LD = 2 (n0 k 0 x0 )/2 is Rayleigh distance or diffraction length, with n0 the linear refractive index,𝑘0 = 2𝜋/𝜆 the wave vector, 𝜆 the wavelength, x0 is the initial beam width, LNL = (n2 k 0 Im )−1 the self-focusing distance, positive n2 the nonlinear refractive index, 𝑋 = 𝑥/x0 and 𝑍 = 𝑧/LD are the normalized lengths along transversal axis and direction of propagation. When LD /LNL = 1 and n2 is positive, Eq. (2) allows one analytical solution known as a fundamental soliton described by Hyperbolic Secant (Sech) profile called bright spatial soliton. Eq. (2) was solved numerically using the split-step method. The split-step method was introduced by Fisher [23] (also known as split step Fourier method) calculates the propagation of an optical field in dispersive and nonlinear media considering small segments in which one of the two properties is absent [24], and has been widely used to solve the NLSE in optical fibers (see for example [25]). The method is reliable and versatile that allows to study new problems [26-28]. The numerical bright spatial soliton is obtained when the following initial field distribution is used: As (𝑋) = Sech(√2𝑋)

(3)

In order to demonstrate that the numerical solution of Eq. (2) produces quasi-solitons when any initial field is used the on-axis intensity was monitored for 100 propagation lengths. The width of the initial field was carefully increased in small amount until the smallest variations were obtained. As representative examples of the different initial conditions tested, we present the numerical propagation considering the following four functions as initial conditions: 10 𝑋 (4) Asg (𝑋) = exp (− ( ) ) bsg 2

(5)

|𝑋| )) be

(6)

Ae (𝑋) = exp (− (

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𝑋 Ag (𝑋) = exp (− ( ) ) bg

𝑋 As (𝑋) = Sech ( ) bs

(7)

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where Asg represents a Super-Gaussian having width bsg , Ag a Gaussian of width bg , Ae an exponential of width be , and As a Sech function of width bs . The previous functions were chosen because present similar or very different characteristics with respect to the Sech: The Gaussian is similar, while the supper-Gaussian is flat at the center and exponential is sharp at the center. It is important to mention that, to obtain quasi-soliton behavior it is necessary to consider an adequate width that depends on the initial function. In Fig. 1, we plot the evolution of each distribution along a 100 Z, together with its initial and final intensity profile. The adequate width for quasi-soliton propagation for the Super-Gaussian, Gaussian and Exponential was of bsg = 1.16, bg = 1.2 and be = 1.1 , respectively. For Sech a width of bs = 0.7071 produces a perfect soliton. Smaller width values produce a diffracted beam, while higher values produce a beam, with a remarkable intensity oscillatory behavior. Compared with the total power of the fundamental soliton, the supper Gaussian is 1.45 times, while the Gaussian is 1.06 times and for the exponential is 0.78 times. Then the power of the fundamental soliton cannot be used to obtain the adequate width of the initial field distribution that generates quasi-solitons. The criteria used to determine when the propagation produces a quasi-soliton can be understood with the help of the on-axis intensity behavior, see Fig. 2. At the beginning the beam is reshaped and the on-axis intensity can increase (Super Gaussian and Gaussian) or decrease (exponential), but after almost 10Z the beam tends to reach a profile and on-axis intensity that propagates with minimum changes. In the case of the Super-Gaussian distribution, the on-axis intensity reaches an average value of 1.78 with variations less than 1%. In the case of the Gaussian distribution, the on-axis average intensity is 1.12 with variations lower than 6%. Finally, for the Exponential profile, the on-axis average intensity is of 0.6 with variations lower than 3%.

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Fig. 1. Propagation (top) for 100 Z for: a) Super-Gaussian, b) Gaussian and c) Exponential initial field distributions. Intensity profiles in linear (middle) and logarithmic (bottom) scale for: initial (blue-solid line) and final (red-dash dot line) distributions. Sech Test (Green marks).

Fig. 2. On-axis intensity along 100Z for the following initial fields: Super-Gaussian (black, dot line) with 𝐛𝐬𝐠 = 𝟏. 𝟏𝟔, Gaussian (blue, solid line) with 𝐛𝐠 = 𝟏. 𝟐 and Exponential (red, dash line) with 𝐛𝐞 = 𝟏. 𝟏, and Sech as perfect soliton (green, dash-dot line) with 𝐛𝐬 = 𝟎. 𝟕𝟎𝟕𝟏 . As reference, vertical line at 10Z.

As we mention, at the beginning of the propagation, the beam is reshaped. The interesting point is that, after some propagation distance in the local medium, the beam intensity transforms to Hyperbolic Secant profile. To verify this fact, we compare the profile at different propagation distances Z with an intensity obtained by the following expression (we call it Sech-Test): 2

(8)

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2|A0 (𝑍)|2 Is (𝑋, 𝑍) = [|A0 (𝑍)|Sech ( 𝑋)] Pi

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where A0 (Z) is the on-axis amplitude of the propagated beam at the position Z. Pi = +∞ ∫−∞ |A(𝑋, 𝑍 = 0)|2 d𝑋 is the initial power. Note that even that Pi is used to adjust the propagated intensity profile this does not means that all the initial power is contained in it. In Fig. 1 the final profiles are compared with that obtained through Eq.(8), plotted as green marks, both in linear (middle row) and logarithmic (bottom row) scale. We can see that the correspondence between both intensity profiles is very good in the central portion. Eq. (8) is adequate when the beam propagates with the minimum variations, however the square Sech profile is reached always after a few propagated distances even the width is smaller than the adequate to generate quasi-soliton. As conclusion of this part of the study is that symmetric initial field profiles are reshaped into a hyperbolic secant when they are propagated in a local nonlinear media, with the adequate initial width these profiles can propagate as quasi-solitons. 2. Numerical Simulation in Nonlocal medium

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The formation of quasi-solitons through arbitrary initial field distributions is exclusive of nonlinear media with local response? To answer this question, we solve numerically the following equation: ∂A(𝑋, 𝑍) 1 ∂2 A(𝑋, 𝑍) (9) −i = + Δn(I)A(𝑋, 𝑍) ∂𝑍 4 ∂𝑋 2

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where Δn(I) is defined in Eq.(1), which describes materials with nonlocal response. Eq. (9) was solved with different initial field amplitudes and response functions. In particular, the following nonlocal response functions were considered: +∞ 𝑋 𝑋 R s (X) = R 0s Sech ( ) → R 0s = 1/ ∫ Sech ( ) d𝑋 αs α s −∞ |X| 1 R e (X) = R 0e exp (− ) → R oe = αe 2αe 2 𝑋 R g (𝑋) = R 0g exp (− ( ) ) → R og = 1/(√παg ) αg +∞

(10( (11)

(12)

R 0s , R 0e and R 0g are normalization factors to fulfill ∫−∞ R(𝑋)d𝑋 = 1 [3], αs , αe and αg are associated to the degree of nonlocality for Sech, Gaussian and exponential nonlocal responses, respectively. The same initial field distributions used in the local case, here also are used, with different response functions and degrees of nonlocality. First, we are going to present the results obtained with the Sech nonlocal response function given by Eq.(10) and using the

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hyperbolic secant, Eq.(7), as initial field profile. Obtaining that for αs smaller than one and bs = 0.7071 the initial profile is slightly affected during the propagation and its solitonic behavior is kept. Degrees of nonlocality equal or larger than 1 required of increasing of the width of the initial profile in order to get the quasi-soliton behavior, therefore we can say that the initial power must be increased as the degree of nonlocality. In Fig. 3 the propagation, initial and final intensity profiles for this field distribution and response function are plotted for different degrees of nonlocality. When αs = 1 the quasi-soliton behavior was obtained when bs = 0.87. The on-axis intensity presented variations smaller than 1.5%. When 𝛼𝑠 = 5 , the adequate width for quasi-soliton behavior was of bs = 1.6 . In this case the on-axis variations are smaller than 16%. When αs = 10 the adequate quasi-soliton width was bs = 2.19, with variations in the on-axis intensity of 18%. Then as the degree of nonlocality increases the variation in the on-axis intensity increases and the beam propagates as an oscillating soliton, this type of behavior is also called breather.

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Fig. 3. Numerical propagation (top) and intensity profiles in linear (middle) and logarithmic (bottom) scale for a hyperbolic secant initial field in a media with Sech nonlocal response and nonlocal degrees αs of: a) 1, b) 5 and c) 10. Input intensity profile (blue), Final intensity profile (red) and Gaussian test (green marks).

In Fig. 4 we plot the initial power as a function of the degree of non-locality, and we observe that increases as the degree of non-locality in order to obtain the best quasi-soliton. Similar behavior of the critical power was obtained considering the hyperbolic secant initial condition with different response functions, Eq.(11) and Eq.(12), with different degrees of nonlocality. When the exponential response function, given by Eq.(11), is considered, smaller initial

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powers are needed. Intermediate values of initial power are needed when a Gaussian response function is considered.

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Fig. 4. Initial power to obtain quasi-soliton propagation when a Sech initial field distribution is used in media with different degrees of nonlocality and nonlocal response function of: Sech (blue, dotted line), Gaussian (red, dashed line) and Exponential (Green, solid line).

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The behavior of the initial power when other initial field distributions were used is plotted in Fig. 5 and Fig. 6. We can observe that, as in the previous case, the same tendency with the degree of nonlocality and the response function is obtained. The most important difference is that the exponential response function requires the smallest initial powers to generate the quasi-soliton behavior while the Gaussian response function requires the highest. This fact can be due to the spatial extension of the different response functions for the same value of α, see Fig. 7. In Fig. 7 the three nonlocal response functions are plotted for αs = αe = αg = 5, where we can observe that the Gaussian nonlocal response is defining the lowest spatial extension, while Sech and Exp have the largest spatial extension.

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Fig. 5. Initial power to obtain quasi-soliton propagation when a Gaussian initial field distribution is used in media with different degree of nonlocality and different nonlocal response function of: Sech (blue, dotted line), Gaussian (red, dashed line) and Exponential (green, solid line).

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Fig. 6. Initial power to obtain quasi-soliton propagation when an exponential initial field distribution is used in media with different degrees of nonlocality and nonlocal response function of: Sech (blue dotted line), Gaussian (red, dashed line) and Exponential (green, solid line).

Fig. 7. Amplitude of the following nonlocal response function: Sech (blue), Exponential (green) and Gaussian (red) for 𝛂 = 𝟓

Another interesting point is that, for all initial field distributions propagating in nonlocal media, described by any nonlocal response function and degree of nonlocality, after some propagation distance the beam intensity profile is reshaped to a Gaussian. This Gaussian test can be obtained from the on-axis amplitude, A0 (𝑍), and the initial power, Pi , using the following expression:

2

2

π |A0 (𝑍)|2 IG (𝑋, 𝑍) = [|A0 (𝑍)| ∗ exp (− (√ 𝑋) ) ] 2 Pi

(13)

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To demonstrate this, in Fig. 3, the final propagated intensity profiles for different degrees of nonlocality, when a sech initial field profile and sech response function are considered, are plotted in a linear and logarithmic intensity scale and compared with that obtained from Eq.(13) (green marks). Notice that a good match between them is obtained in the central part of the profiles. As in the local case, to use Pi does not mean that all the initial power is contained in the part of propagated profile that was compared with the Gaussian test.

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In order to demonstrate that the same profile is acquired after propagation in nonlocal media, in Fig. 8, as example, are plotted the propagation and intensity profiles when a Gaussian and an Exponential initial field distributions are propagated in a medium with a Sech response function with αs = 5 . Note that in the case of the Gaussian initial condition the beam propagation presents very small variations, almost as a perfect soliton. The conclusion of this part of the study is that any initial field distribution propagated in a nonlocal nonlinear media, with arbitrary degree of nonlocality and response function, after some propagation distance the intensity profile tends to be modified to a Gaussian. With the adequate initial width in some cases the propagated profile generates a quasi-soliton.

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Fig. 8. Numerical propagation (top) and intensity profiles in linear (middle) and Logarithm (bottom) intensity scale for Gaussian (a) and Exponential (b) initial field distribution propagated in a media with Sech nonlocal response function for 𝛂 = 𝟓 . Initial profile (blue, solid line), final profile (red, dash line) and Gaussian test (green marks).

3. Conclusions In this paper we have studied the numerical propagation of arbitrary initial field distributions in positive 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

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initial beam profiles far from hyperbolic secant distribution, but with an adequate beam-width, can propagates as a soliton or more precisely a quasi-soliton, since small oscillations can be detected. As evidence two extreme initial profiles were presented: Super-Gaussian (as a very wide beam) and Exponential (with sharp central amplitude). After propagation all the initial arbitrary profiles evolve to a hyperbolic secant intensity profile. A formula to obtain the propagated profile was given. Furthermore, the numerical propagation of arbitrary initial field profiles was analyzed in nonlocal media, with particular response functions and any degree of nonlocality. The behavior observed was similar to the local case: the arbitrary initial beam profiles, with the adequate width, can propagate as a quasi-soliton after some propagation distance. In the nonlocal case, independently of the response functions of the media and the initial field considered, the intensity profile tends to acquire a Gaussian distribution. A formula for the propagated intensity profile was given. The behavior reported can be extended to asymmetric initial distributions that give rise to one beam. For negative nonlinear materials, it is necessary to do the analysis.

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Acknowledgments Majid Hesami, thanks CONACYT Grant 716213. Work partially supported by CONACYT Mexico under the project A1-S-23120

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