Modulation instability conditions for the generation of femtosecond pulses represented by perturbative nonlinear Schrödinger equation with higher order dispersion and nonlinear effects is investigated. It is observed that modulation instability occurs even in the normal dispersion regime due to the presence of fourth order dispersion

Kalithasan, B.

Porsezian, K.

English

Publications of the IAS Fellows

NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS 1
Modulation Instability of Perturbative Nonlinear Schro¨dinger
Equation with Higher Order Dispersion
B.Kalithasan and K.Porsezian
Abstract— Modulation instability conditions for the generation of fem-
tosecond pulses represented by perturbative nonlinear Schro¨dinger equation
with higher order dispersion and nonlinear effects is investigated. It is ob-
served that modulation instability occurs even in the normal dispersion regime
due to the presence of fourth order dispersion.
I. INTRODUCTION
Extensive research has been carried out in the field of soliton
propagation in optical fibers [1-3]. In the picosecond regime,
the nonlinear Schro¨dinger equation describes the bright / dark
solitons in the anomalous / normal dispersion regime and the
plane wave shows the modulation instability. In the case of
dispersion shifted fibers operating in the femtosecond regime,
the constant guroup velocity dispersion term becomes relatively
small and as a result, higher order corrections are called for. The
major higher order terms come from the dispersion slope, the
dispersive effect of the Kerr coefficient and the self-induced Ra-
man scattering (SRS). Taking into account of these higher order
terms the nonlinear Schro¨dinger equation is perturbative to the
higher order nonlinear Schro¨dinger equation (HNLSE) [2-5]. In
the case of optical solitons, when femtosecond pulses are con-
cerned, the effect of nonlinear dispersion is so significant that
it can no longer be treated as a perturbation term. Perturbative
nonlinear Schro¨dinger equation has been integrated by using the
lie transform technique [6]. The effect of nonlinear dispersion
term can be studied exactly while the effect of other perturba-
tion terms has to be studied with the aid of soliton perturbation
theory. This was done by Shchesnovich and Doktorov based on
the Riemann-Hillbert problem [7] .
On par with the soliton propagation, continuous wave prop-
agation in optical fibers have also demanded special attention
[1-3]. A continuous wave with a cubic nonlinearity in an anoma-
lous dispersion regime is known to develop instability with re-
spect to small modulations in amplitude or in phase in the pres-
ence of noise or any other weak perturbation, called modulation
instability (MI) [1-3]. Generally, the perturbation has its ori-
gin from quantum noise or from a frequency shifted signal wave
[3]. The MI phenomenon was discovered in fluids [8], in non-
linear optics [9] and in plasmas [10]. MI of a light wave in an
optical fiber was suggested by Hasegawa and Brinkman [11] as
a means to generate a far infrared light source and since then
has attracted extensive attention for both its fundamental and
applied interests [8, 9]. As regards applications, MI provides a
natural means of generating ultra short pulses at ultra-high repe-
tition rates and is thus potentially useful for the development of
high speed optical communication systems in future and hence
B.Kalithasan and K.Porsezian (corresponding author) are with Department
of Physics, Pondicherry University, Pondicherry 605 014. e-mail: ponz-
sol@yahoo.com. KP wishes to thank the CSIR, DST, UGC (Research Award)
and DAE-BRNS, Government of India, for the financial support through
projects.
has been exploited a great deal in many theoretical and experi-
mental studies for the realization of laser sources adapted to ul-
trahigh bit-rate optical transmissions [11, 12]. When breakup of
continuous wave and quasi-continuous wave radiates into a train
of picosecond and femtosecond pulses in the fiber, higher-order
nonlinear effects such as self-steepening (SS), self-induced Ra-
man scattering (SRS) and higher order dispersion effects such
as third and fourth order dispersion should also be taken into
account [1-3]. The influence of self-phase modulation (SPM),
higher order nonlinear effects and higher order dispersion ef-
fects on MI in the anomalous dispersion regime have been ob-
tained in Ref. [1-3] and the following conclusion have been ar-
rived at: The instability conditions that govern the generation of
ultrashort pulses in the anomalous dispersion regime are not af-
fected irrespective of the presence or absence of the dimension-
less third order dispersion coefficient. The effect of SRS on MI
is such that for comparatively small values of the perturbation
frequency, group velocity dispersion and self-phase modulation
terms dominate whereas for comparatively large values the per-
turbation frequency, the gain spectrum increases linearly with
the result that the region of MI is widened due to SRS. More-
over the self-steepening effect reduces the maximum gain and
bandwidth. This article purports to the investigation of MI con-
dition for the generation of femtosecond pulses represented by
perturbative HNLSE. The remainder of the article is arranged as
follows: In sec. II, the theoretical model is discussed. Linear
stability analysis of the governing equation is performed in Sec.
III. Results and discussions are discussed in Sec. IV. Section V
comprises the the conclusion.
II. THEORETICAL MODEL
The governing equation depicting femtosecond soliton prop-
agation in nonlinear optical fibers with higher order linear ef-
fects such as third and fourth order dispersion effects and higher
order nonlinear effects such as SS, SRS, nonlinear gain/loss
etc., is represented by the perturbative higher order nonlinear
Schro¨dinger equation and is given by [14]
iqz −
ρ
2
qtt + α|q|
2q = iε(λ(|q|2q)t + νq(|q|
2)t)− γqttt
+iqtttt − iβ(q
2q∗t )t − iΓq
2
t q
∗ − iηq∗q2tt + δ|q|
2mq), (1)
where q is the slowly varying envelope, z is the propagation di-
rection, t is the retarded time, ρ is the group velocity dispersion
(GVD) coefficient, α is the self-phase modulation (SPM) coef-
ficient and ε is the width of the spectrum. The right hand side
of the governing equation comprises of the perturbative terms
where λ is the self steepening coefficient, ν is the self-induced
Raman scattering (SRS) coefficient, γ is the third order disper-
sion (TOD) coefficient, σ is the fourth order dispersion (FOD)
coefficient. The terms β,Γ and η arise in the context of quasi-
solitons in fiber optics [14,15] and δ is the nonlinear damping
2 NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
or amplification depending on its sign and m could be 0, 1, 2.
The nonlinear damping term δ, m = 0, is commonly known as
attenuation, while for m = 1 or 2 we get the cubic or the quintic
saturation, respectively. Here, we have considered the case for
m = 0.
Using the following transform given by
q(z, t) = Q(z, t)eεδz, (2)
into equation (1), we get
iQz −
ρ
2
Qtt + αf |Q|
2Q = iε(f(λ(|Q|2Q)t + νQ(|Q|
2)t)
−γQttt + iQtttt − if(β(Q
2Q∗t )t + ΓQ
2
tQ
∗ + ηQ∗Q2tt)), (3)
where f = exp(2εδz).
We consider equation (3) as the governing equation and ob-
tain the MI condition for the generation of femtosecond pulses.
III. LINEAR STABILITY ANALYSIS
Steady state solution of equation (3) can be written as
Q(z, t) =
√
P0e
iφ(z), (4)
where P0 is the input power and φ(z) is the nonlinear phase
shift. Substituting equation (4) into equation (3), we get
φ(z) = αP0
(e2εδz − 1)
2εδ
. (5)
The linear-stability of the steady state can be examined by
introducing a perturbed field of the form
Q(z, t) = (
√
P0 + a(z, t))e
iφ(z), (6)
where P0 is the incident pump power and |a(z, t)|2 << P0.
We assume for the perturbation a(z, t) the following ansatz with
frequency detuning from the pump Ω:
a(z, t) = u(z)e−iΩt + v(z)eiΩt, (7)
where u(z) and v(z) are the complex perturbation fields. By
substituting equation (6) into equation (3) and collecting the lin-
ear terms in u(z) and v(z), we obtain the equation for the per-
turbed field as
i
du(z)
dz
=M11u(z) +M12v
∗(z), (8)
i
dv∗(z)
dz
=M21u(z) +M22v
∗(z), (9)
where
M11 = e
2εδzP0(−α+ 2ελΩ+ ενΩ− 2εηΩ
2)
−
ρΩ2
2
+ εγΩ3 − εσΩ4,
M12 = e
2εδzP0(−α+ ελΩ+ ενΩ− εβΩ
2),
M21 = e
2εδzP0(α+ ελΩ+ ενΩ+ εβΩ
2),
M22 = e
2εδzP0(α+ 2ελΩ+ ενΩ+ 2εηΩ
2)
+
ρΩ2
2
+ εγΩ3 + εσΩ4. (10)
Equations (8) and (9) can be written as
i
d
dz
[
u(z)
v∗(z)
]
=

 M11 M12
M21 M22


[
u(z)
v∗(z)
]
. (11)
The eigenvalues of the matrix M determine the wave num-
ber K of the perturbation. The modulation instability occurs
when K possesses a nonzero imaginary part. The eigenvalues
are given by dispersion relation
K =
1
2
(M11 +M22 +
√
(M11 −M22)2 + 4M12M21 ). (12)
The importance of modulation instability is measured by a
power gain defined by
G(Ω) = 2|Im(K)|, (13)
where Im(K) represents the imaginary part of K.
IV. RESULT AND DISCUSSION
We have observed that the TOD order dispersion term has no
influence on the MI condition but the second order dispersion
and FOD terms have influence in the MI condition.The MI con-
dition. MI gain increases due to the influence of the SRS effect.
From the MI gain spectrum, space between the two sidebands
is more when compared with zero and anomalous dispersion
regimes.
-50
0
50 0
0.1
0.2
0
20
40
60
Ω
P
G(Ω)
Fig. 1: MI Gain spectrum in the normal dispersion region for various values
of ρ = 0.0114, α = 0.7, γ = 1.18 × 10−4, σ = −6.44 × 10−7, δ =
−0.04, ε = 1, λ = 0.63, β = 0.04, z = 0.2, η = 0.03, ν = 0.07.
figure
RIASM, UNIVERSITY OF MADRAS, CHENNAI, 6-8 FEB. 2006 3
-50
0
50 0
0.1
0.2
0
20
40
Ω
P
G(Ω)
Fig. 2: MI Gain spectrum in the zero dispersion region for various values of
ρ = 0, α = 0.7, γ = 1.18× 10−4, σ = −6.44× 10−7, δ = −0.04, ε =
1, λ = 0.63, β = 0.04, z = 0.2, η = 0.03, ν = 0.07.
figure
-50
0
50 0
0.1
0.2
0
10
20
30
Ω
P
G(Ω)
Fig. 3: MI Gain spectrum in the anomalous dispersion region for various val-
ues of ρ = −0.0114, α = 0.7, γ = 1.18× 10−4, σ = −6.44× 10−7, δ =
−0.04, ε = 1, λ = 0.63, β = 0.04, z = 0.2, η = 0.03, ν = 0.07.
figure
V. CONCLUSION
We have investigated the modulation instability of perturba-
tive nonlinear Schro¨dinger equation with higher order disper-
sion in normal, anomalous and zero dispersion regimes. We
have observed that while TOD dispersion term doesn’t influence
the MI, second order dispersion and FOD terms do influence.
The MI is dependant on the signs of the second order dispersion
and FOD terms. In normal dispersion regime, MI occurs only
for negative values of the higher order dispersion coefficient. We
have observed the maximum gain in normal dispersion regime
when compared with zero and anomalous dispersion regimes.
REFERENCES
[1] A. Hasegawa in Optical Solitons-Theoretical and Experimental Chal-
lenges (Eds. K. Porsezian and V. C. Kuriakose, Springer Verlag, Berlin,
2003).
[2] G. P. Agrawal, ’Nonlinear Fiber Optics’, Academic Press, San Diego
(2001).
[3] G. P. Agrawal, ’Applications of Nonlinear Fiber Optics’, Second Edition,
Academic Press, New York (2001).
[4] Y. S. Kivshar and G. P. Agrawal, ”Optical solitons -From fibers to pho-
tonic crystals”, Academic Press, New York (2003).
[5] A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers, Springer
Verlag, Germany (2003).
[6] A. Hasegava and Y. Kodama, Solitons in Optical Communica-
tions,Oxford University Press (1995).
[7] V.S. Shchesnovich and E.V. Doktorov, Physica D. 129, 115 (1998).
[8] M. Nakazawa, K. Suzuki, H. Kubota and H. A. Haus , Phys. Rev. A39,
5768 (1989)
[9] L. A. Ostrovskii, Sov. Phys. JETP 24, 797 (1967).
[10] V. I. Karpman, JETP Lett. 6, 277 (1967).
[11] A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. 16, 694
(1980).
[12] G. Millot, P. Tchofo Dinda, E. Seve, and S. Wabnitz, Optical Fiber Tech-
nology 7, 170 (2001).
[13] G. Millot, S. Pitois, J. M. Dudley and M. Haelterman in Optical Solitons-
Theoretical and Experimental Challenges (Eds. K. Porsezian and V. C.
Kuriakose, Springer Verlag, Berlin, 2003).
[14] A. Biswas, Journal of Electromagnetic Waves and Applications 14, 95
(2000).
[15] V.E. Zakharov,E.A. Kuznetsov, Journal of Experimental and Theoretical
Physics 86, 1035 (1998).
[16] F.Kh. Abdullaev, S.A. Darmanyan, S. Bischoff, P.L. Christiansen and
M.P. Sorensem, Opt. Comm. 108, 60 (1994).
[17] S.B. Cavalcanti, J.C. Cressoni, H.R. da Cruz, A.S. Gouveia-Neto,
Phys.Rev. A 43, 6162 (1991).
[18] F. Ito, K.-i. Kitayama, H. Yoshinaga, Appl.Phys.Lett. 54, 2503 (1989).


Modulational instability of perturbative nonlinear Schrödinger equation with higher order dispersion

http://repository.ias.ac.in/83962/1/88-pub.pdf

Modulational instability of perturbative nonlinear Schrödinger equation with higher order dispersion

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows