Dynamically Stable Super Gaussian Solitons in Semiconductor Doped Glass Fibers

J. of Electromagn. Waves and Appl., Vol. 20, No. 7, 901–912, 2006

DYNAMICALLY STABLE SUPER GAUSSIANSOLITONS IN SEMICONDUCTOR DOPED GLASSFIBERS

Shwetanshumala

A. N. CollegePatna-800013, India

A. Biswas

Department of Applied Mathematics and Theoretical PhysicsDelaware State UniversityDE 19901-2277, USA

S. Konar

Department of Applied PhysicsBirla Institute of TechnologyMesra-835215, Ranchi, India

Abstract—In this paper we have investigated the propagation prop-erties of chirped super Gaussian soliton pulses through semiconductordoped glass fibers which is a dispersive medium with cubic quinticnonlinearity. Using variational principle, the evolution equations ofthe soliton parameters such as amplitude, temporal width, position ofcentre, chirp etc. have been derived. The dynamics of these parame-ters have been analysed. Employing linear stability analysis we haveshown that super Gaussian solitons are stable in semiconductor dopedglass fibers.

1. INTRODUCTION

Propagation of optical solitons in dispersive fibers has been afascinating area of extensive theoretical and experimental studies.Hasegawa and Tappert [1] were the first to suggest the possibilityof optical solitons in fibers, while Mollenauer et al. [2] were the

902 Shwetanshumala, Biswas, and Konar

first to experimentally demonstrate the existence of such solitonsin optical fibers. Their remarkable stability and persistence ofpropagation characteristics have been found to have tremendous edgeover electronic information carrier leading to the proposal of opticalsoliton based communication systems [3–6]. Optical solitons havebeen studied extensively because of their potential applications in longdistance communication and all optical ultra fast switching devices[7, 8].

The space-time evolution of a soliton is described by nonlinearSchrodinger equation (NLSE). The NLSE includes dynamic interplaybetween group velocity dispersion (GVD) and nonlinearity inducedself phase modulation (SPM). GVD causes temporal broadening aswell as chirping, while the primary effect of SPM is the generation ofchirp keeping temporal shape unchanged. The NLSE is completelyintegrable which has single and higher order multisoliton solutions.Single solitons propagate without changes in their shapes [4–7]and they are robust against small perturbations. The balance ofnonlinearity and dispersion is satisfied only over a small length ofthe fiber because of the fiber loss. As a result the essential balancebetween GVD and SPM is disturbed. Large number of theoretical andexperimental investigations have been devoted to tackle the problem ofsoliton stability in a long monomode optical fiber. Tajima [8] suggestedthat with an axially inhomogeneous fiber with a cone tapered core thatcan control GVD, a non-dispersive soliton can be obtained. Use ofdispersion compensated optical fibers with loss or use of periodicallyvarying core diameter to control soliton parameters can also be helpful.

Most of the earlier works on soliton propagation are confined tothe Kerr medium characterised by small value of nonlinear coefficient.However, higher order nonlinearities [9–23] become important insemiconductor doped glass fibers and other composite materials atmoderate intensities. Thus pulse propagation in fibers made of suchmaterials needs to be described with different forms of nonlinearityin place of usual Kerr nonlinearity. One important model whichdescribes the soliton propagation in semiconductor doped glass fibersis the cubic quintic nonlinearity with nonlinear polarisation (P ) andrefractive index (n) given respectively [24] by following equations

P =34ε0χ

(3)|E|2 +58ε0χ

(5)|E|5 (1)

n = n0 + n2|E|2 + n4|E|4 (2)

where, E being the electric field, n0 is the linear refractive index ofthe medium, n2, n4 are respectively the third and fifth order nonlinearrefractive index coefficients and χ(3), χ(5) are the third and fifth order

Dynamically stable super gaussian solitons 903

susceptibilities respectively. Refractive index and hence values ofχ(3) and χ(5) are controlled by doping semiconductor fibers with twodopants of properly chosen characteristics. Suitable positive value ofχ(3) and negative value of χ(5) opens the door of new possibilities,hence has attracted a large number of investigators [25, 26]. Thusalong this line, we have investigated the propagation characteristicsof a chirped super Gaussian (SG) soliton in semiconductor dopedglass fibers (SDG) having nonlinearity up to fifth order. The paper isorganised in three sections. Section 2 contains the equation governingsoliton propagation. Variational formalism has been evoked to derivethe coupled ordinary differential equations (ODE), which have beenused subsequently to study soliton propagation. In Section 3, linearstability analysis has been employed to investigate stationary solutionsand robustness of these solitons with results and discussion. Finally, abrief conclusion has been added in Section 4.

2. NONLINEAR PROPAGATION

We consider an optical pulse propagating through a semiconductordoped dispersive fiber possessing cubic quintic nonlinearity. Thepropagation of the slowly varying pulse-envelope u along z direction isgoverned by the nonlinear Schrodinger equation [24]

i∂u

∂z+

12∂2u

∂t2+ |u|2u+ γ|u|4u = 0 (3)

where γ(= kχ(5)/χ(3), k is some constant) includes the contribution offifth order nonlinearity of the medium, and pulse envelope u is obtainedafter suitable normalization which can be found elsewhere [24]. Ingeneral fifth order nonlinearity is small in comparison to the thirdorder i.e., Kerr nonlinearity. However, employing doping technique,the value of this nonlinearity can be enhanced in semiconductor dopedglass fibers. In spite of that, the ratio of fifth and third order nonlinearsusceptibilities is much small in comparison to unity. However, ina real experiment the case of self defocusing quintic nonlinearity isonly interesting and has drawn considerable attention. For such realexperimental situation [6, 18, 29] γ is always less than 1, and typicalvalue of γ ranges from |γ| ≈ 0.001 to 0.1. In our present investigationwe shall confine our attention in the above mentioned regime i.e.,0 ≺ γ ≺ 1. Equation (3) is written in a frame of reference in whichthe pulse is moving with the group velocity. The NLSE representedby equation (3) has an infinite number of symmetries with eachsymmetry corresponding to a conserved quantity. Two most important


conserved quantities are energy K, also known as pulse power andlinear momentum M of the soliton given by

K =+∞∫

−∞|u|2dt and (4)

M =i

2

+∞∫−∞

(u∂u∗

∂t− u∗∂u

∂t

)dt (5)

The equation (3) is known as modified nonlinear Schrodinger equationwhich can be reformulated as a variational problem [27, 28] as follows

δ

∫L

(u, u∗,

∂u

∂z,∂u∗

∂z,∂u

∂t,∂u∗

∂t, z

)dt = 0 (6)

where, the Lagrangian density L is given by

L =i

2

(u∗∂u

∂z− u∂u

∗

∂z

)− 1

2

∣∣∣∣∂u∂t∣∣∣∣2

+|u|42

+γ|u|6

3(7)

The soliton solution of Eq. (3) may be assumed to be a chirped pulseof the form

u(z, t) = A(z)f [B(z){t− t0(z)}]× exp

[iC(z)(t− t0)2 − iκ(z){t− t0(z)} + iθ0(z)

](8)

where f represents the shape of the soliton, A(z), B(z), C(z), κ(z), t0(z)and θ0(z) represent the amplitude, inverse width, chirp, nonlinear fre-quency shift, the centre, and the phase of the soliton respectively. Theaverage Lagrangian 〈L〉 of the system is obtained by integrating equa-tion (7) w.r.t. time with the trial function given by (8). Thus,

〈L〉 =+∞∫

−∞Ldt (9)

〈L〉 = −A2

(B

2I002 + 2

C2

B3I220 +

κ2

2BI020

)+A4

2BI040

−A2

B3I220

dC

dZ+A2

BI020

(κdt0dz

+dθ0dz

)+γA6

3BI060 (10)

where Iabc =+∞∫−∞τaf b(τ)

(dfdτ

)cdτ and

τ = B(z){t− t0(z)} (11)


By deriving Euler-Lagrange’s equation for the six soliton parametersA, B, C, κ, t0 and θ0, we get the set of following coupled ODE’s.

dA

dZ= −AC (12a)

dB

dZ= −2BC (12b)

dC

dZ=

(B4

2I002I220

− 2C2

)− A

2B2

4I040I220

− γA4B2

3I060I220

(12c)

dκ

dz= 0 (12d)

dt0dz

= −κ (12e)

dθ0dz

=

(κ2

2− I022I020

B2

)+

5A2

4I040I020

+4γA4

3I060I020

(12f)

From (12a) and (12b) we observe that A = N√B for some constant N .

Leaving out the redundant equations we get two evolution equationsas follows

dB

dz= −2BC (13)

dC

dz=

(B4

2I002I220

− 2C2

)− N

2B3

4I040I220

− γN4B4

3I060I220

(14)

For SG pulse we choose

f(τ) = e−(τ2)m(15)

The Super Gaussian parameter ‘m’ controls the degree of edgesharpness. m = 1 corresponds to the case of a Gaussian pulse, whichgradually becomes bell shaped with sharper leading and trailing edgeswith increasing m. Pulse forms for different m values has been givenin Fig. 1. With SG pulse of the form (15), pulse power K and linearmomentum M turn out to be

K = N2I1 (16)

M = = −N2κI1 (17)


-2 -1 0 1 20

0.5

1

τ

m=1

m=2

m=3

m=4

f(τ)

Figure 1. Typical profile of supergaussian solitons with differentsupergaussian parameters.

3. RESULTS AND DISCUSSION

From equations (13) and (14) a nontrivial stationary point (Bs, Cs) forSG pulses is given by

Cs = 0 (18)

Bs =32

N2I040(3I002 − 2γN4I060)

(19)

We have solved equation (19) to get stationary values of B for variousvalues of N and the parameter m. Such values of B for which pulseremains stationary are displayed in Fig. 2. With initial values of B andC chosen for a stationary state, we observe from numerical simulationthat, in fact B and C remains constant with distance of propagation,which means pulse propagates such that its width and chirp remainthe same. This has been depicted in Fig. 3.

In order to see whether the SG pulse is stable, we undertake linearstability analysis about the stationary point (Bs, Cs). We introducesmall perturbation to the values corresponding to the equilibriumpoints and take B = Bs + ∆B, C = Cs + ∆C, where ∆B and ∆C arevery small in comparison with their respective stationary values. Afterlinearising equations (13) and (14) around stationary point, we get aset of two equations, which can be put, in a matrix form

d

dz

(∆C∆B

)=

(0 J12

J21 0

) (∆C∆B

)(20)


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

W

1 2 3 4 5 6 7N

Figure 2. Variation of inverse width (W = 1/B) of stationarysupergaussian solitons for different supergaussian parameters. Solidline m = 1.0, dashed line m = 1.5 and dotted line m = 2.0.

0 0.5 1 1.5 2 2.5 3-0.5

0.5

1.5

B,C

0 0.5 1 1.5 2 2.5 3-0.5

0.5

1.5

B,C

0 0.5 1 1.5 2 2.5 3-0.5

0.5

1.5

B,C

Z

m=1

m=1.5

m=2

(a)

(b)

(c)

Figure 3. Depicting propagation of SG pulse in doped fiber withstationary values of B and C (B = Bs and C = Cs). Solid line B,broken line C. N = 2, γ = −0.01. (a) Bs = 1.33, Cs = 0 (b)Bs = 1.24, Cs = 0 (c) Bs = 1.11, Cs = 0.


0 10 20 30-0.5

0.5

1.5

B,C

0 5 10 15 20 25 30-0.5

0.5

1.5 B

,C

0 5 10 15 20 25 30-0.5

0.5

1.5

B,C

Z

m=1

m=1.5

m=2

(a)

(b)

(c)

Figure 4. Variation of B and C with distance of propagation whenonly one initial value i.e., value of is different form stationary valueBs. B(0) = Bs + ∆, C(0) = Cs = 0 (a) B(0) = 1.33 + ∆, (b)B(0) = 1.24 + ∆, (c) B(0) = 1.11 + ∆, ∆ = 0.2. Solid line B, brokenline C. N = 2, γ = −0.01.

where J12 = 916 · N6I3

040

(3I002−2γN4I060)2and

J21 = − 3N2I040(3I002 − 2γN4I060)

(21)

The eigen values of the 2× 2 stability matrix satisfy the characteristicequation

λ2 +2716

· N6I4040(3I002 − 2γN4I060)

3 = 0 (22)

The two eigen values λj(j = 1, 2) of the 2 × 2 stability matrix are

λ1,2 = ±√

∆ (23)

where

∆ = −2716

· N6I4040(3I002 − 2γN4I060)

3 (24)

For a SDG fiber γ has a small negative value so that ∆ is negativeon all the three curves of Fig. 2. Thus the two eigen values λ1 and


0 10 20 30-0.5

0.5

1.5

B,C

0 10 20 30-0.5

0.5

1.5 B

,C

0 10 20 30-0.5

0.5

1.5

B,C

m=1

m=1.5

m=2

Z

(a)

(b)

(c)

Figure 5. Variation of B and C with distance of propagation whenonly one initial value i.e., value of is different form stationary valueBs. B(0) = Bs + ∆, C(0) = Cs = 0 (a) B(0) = 1.33 + ∆, (b)B(0) = 1.24 + ∆, (c) B(0) = 1.11 + ∆, ∆ = 0.2. Solid line B, brokenline C. N = 2, γ = −0.01.

0.5 1 1.5 2-1

0

1

C

B

m=1

m=1.5

m=2

N=2 γ=-0.01

Figure 6. Phase trajectory of solitons for different supergaussianparameters (m). Closed trajectory signifies solitons are stable againstsmall perturbation.


λ2 are purely imaginary which implies that the system is in oscillatingequilibrium or neutral equilibrium. Small perturbation causes the pulseparameters oscillate about the stationary value. If width of the pulseis changed by a small amount from its stationary value, we observethat pulse width B and chirp oscillate with finite amplitude as thepulse propagates. This has been depicted in Fig. 4. The variation of Band C with the distance of propagation when both have initial valuesslightly different from stationary value Bs and Cs is depicted in Fig. 5.In both cases the oscillation appears to be simple harmonic implyingstable propagation against small perturbation. Fig. 6 depicts the phasediagram of pulse width and chirp when the system is subject to smallperturbation about the stationary point. The trajectories are closedwhich means that the pulse is stable against small perturbation.

4. CONCLUSION

We have studied the propagation of a Super Gaussian soliton in along SDG fiber in which the nonlinearity is cubic quintic. Employingvariational formalism we have constructed a field Lagrangian for thesoliton with six slowly varying free parameters describing the solitonamplitude, inverse width, chirp, frequency, centre of the soliton andthe centre of the phase of the soliton. The dynamics of the soliton isdescribed by a set of coupled ODE for the free parameters. In losslessmedium we get exact analytic solutions for the behaviour of the width,frequency, chirp etc. Using the technique of linear stability analysis,we are able to determine conditions under which soliton propagationmight be possible without distortion. The behaviour of the analyticallyobtained soliton has also been compared by numerically solving theNLSE, a good agreement is obtained.

ACKNOWLEDGMENT

One of the authors S. Konar would like to thank Prof. H. C. Pande,Prof. S. K. Mukherjee and Prof. P. K. Barhai for constant support andencouragement. Swetanshumala acknowledges helpful discussion withMr. S. Jana.

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Dynamically Stable Super Gaussian Solitons in Semiconductor Doped Glass Fibers

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