ISSN 1060�992X, Optical Memory and Neural Networks (Information Optics), 2011, Vol. 20, No. 4, pp. 237–246. © Allerton Press, Inc., 2011.

237

1. INTRODUCTION

For generation of spectral continuum by a laser pulse propagating in a nonlinear medium, the basicpoint is the presence of laser intensity�dependent term in the index of refraction of the medium with Kerrnonlinearity. Appearance of the intensity dependent term leads in case of short laser pulses to physicallymeaningful modulation of the laser pulse phase, i.e., self�phases modulation (SPM). Within the frames ofthe approximation taking into account only the first order of material dispersion the temporal self�actionof laser pulse leads to symmetric broadening of its spectrum. However, a number of physical mechanismslead to asymmetry of spectral broadening even at moderate intensities of laser pulse.

Three most important mechanisms are associated with spatial self�action, formation of shock�wavefront of pulse envelope, and finite time of nonlinear response of the medium. Spatial self�action is asso�ciated with the Kerr nonlinearity of the medium [1]. The formation of the shock wave is caused by theintensity dependence of the pulse group velocity [1]. Effects connected with the finite time of nonlinearresponse of the medium become especially considerable for few�cycle pulses when the medium nonlin�earity can no longer be taken as instantaneous [1]. The delay of nonlinear response is equivalent to disper�sion of nonlinear medium in frequency representation. Laser pulse propagating in a medium with delayingnonlinearity suffers low�frequency shift. Spectral broadening induced by the delayed nonlinearity is thusequivalent to stimulated Raman scattering (SRS).

At excitation of Raman resonance by a few�cycle laser pulse the field spectral component shifted by thefrequency of molecular vibrations, i.e., the Stokes component, is already contained in the pulse spectrum.The SRS process has in this case the character of some kind Raman self�action. At the expense of excita�tion of molecular vibrations energy redistribution occurs in the pulse spectrum and this leads to a red shiftof the spectrum peak [1]. During propagation of an extremely short laser pulse, few�cycle pulse, in an iso�tropic nonlinear medium, if the central wavelength of the pulse is in the range of zero dispersion of themedium, dispersion effects at the early stage of propagation will be determined by higher�order (cubic andhigher) terms in the expansion of the index of refraction. However, during further propagation shift of thecentral wavelength occurs because of spectral broadening caused by both non�inertial and inertial nonlin�

Computer Simulation of the Propagation of Extremely Short Femtosecond Pulse in Single�Mode Fiber

with Shifted Dispersion CharacteristicsD. L. Hovhannisyana, V. O. Chaltykyanb, A. O. Hovhannisyana,

G. D. Hovhannisyana, and K. A. Hovhannisyana

aYerevan State University, ArmeniabInstitute for Physical Research, NAS of Armenia

e�mail: hovhan_artur@yahoo.comReceived in final form, May 3, 2011

Abstract—We present the results of theoretical study of generation of spectral supercontinuum by anextremely short femtosecond laser pulse with the central wavelength corresponding to zero dispersion,propagating in a single�mode fiber with shifted dispersion characteristics. High�order nonlinearSchrödinger equation (HONSE) is considered with higher�order terms taken into account. Numeri�cal integration of HONSE is performed by the method of lines. Time dependence of instantaneous fre�quencies of the femtosecond pulse is obtained. The evolution of the pulse spectrum is calculated andthe medium length dependence of the spectrum width is obtained. For study of the time�frequencydynamics of extremely short femtosecond laser pulse propagating in single�mode optical fiber withallowance for higher order terms, the Wigner transformation (WT) was used.

Keywords: nonlinear Schrödinger equation, method of lines, Raman response, finite difference,Wigner transformation.

DOI: 10.3103/S1060992X11040047

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HOVHANNISYAN et al.

earities of the medium. This means that as the pulse travels in the medium, dispersion spreading is deter�mined by also the second order term in the expansion above. In particular, during propagation in a single�mode fiber of 100 fs pulse with the central wavelength corresponding to zero dispersion the spectral broad�ening is insignificant [2]. But, if a 350 fs pulse travels in an optical fiber with compensated dispersion char�acteristics, an essential spectral broadening takes place [3]. This is caused by the fact that the decrease inthe effective mode area in this case leads to the increase in the effective nonlinearity and mode dispersion.The change in mode dispersion leads in its turn to the shift of the central wavelength, corresponding tozero dispersion, from 1300 to 800 nm, which coincides with the central wavelength of titanium�sapphirefemtosecond laser. The mode dispersion may completely be eliminated, if structural parameters of thefiber will be chosen so that only one, fundamental, mode will travel in the lightguide. Single�mode opticalfibers have this property. But the dispersion may also be shifted by means of quartz glass doping. Suchlightguides are called optical fibers with shifted dispersion; they can have zero dispersion at the wavelengthwith minimum damping (1550 nm). In such fibers the second value is observed of the central wavelengthcorresponding to zero dispersion, which is shifted towards infrared range [4]. These fibers are used for gen�eration of spectral supercontinuum in the infrared range.

For description of this generation process in case of propagation of extremely short femtosecond laserpulse, HONSE may be used where both dispersion terms and higher order nonlinearities are taken intoaccount. Strictly speaking, we must in this case solve the system of Maxwell equations [5, 6]. Nevertheless,if the assumption that the complex amplitude of the pulse varies slowly during the mean period of oscilla�tions (approximation of slowly varying amplitudes (SVA)), HONSE may be used [7–10]. When the con�ditions of applicability of SVA do not met, there exist analytical methods based on the description of evo�lution of the pulse electric field, rather than slow amplitude [11, 12].

In the present work we describe the results of theoretical study of propagation of extremely short fem�tosecond pulse, with the wavelength in the zero�dispersion range, in a single�mode fiber with shifted dis�persion characteristics. We consider HONSE where the fifth�order nonlinearity and the fourth and fifthorder dispersion terms are taken into account. Numerical integration of such HONSE is performed by themethod of lines. Time dependence of instantaneous frequency of the pulse and the evolution of its spec�trum are calculated. We also obtained the medium length dependence of the pulse spectrum width. Thetime�frequency dynamics of propagation is performed with use of Wigner transformation [13].

2. HONSE WITH ALLOWANCE FOR THE FIFTH ORDER NONLINEARITY AND FOURTH AND FIFTH ORDER DISPERSION TERMS

Let us consider the process of propagation of an extremely short femtosecond laser pulse along thez�axis in a single�mode fiber. This process is described by HONSE which can be represented in the form(notations of [1])

(1)

where , A(ξ,τ) is the slowly varying amplitude, P0 the peak power of the pulse,

, β2 is the group velocity dispersion, τ0 the pulse duration determined at half�maximum

THM ≡ = 1.665τ0 (for a Gaussian pulse), , vg is the group velocity, ,

, is the coefficient of nonlinearity, c the speed of light in vacuum, ω0

the pulse carrier frequency, Aeff the effective area of the mode, n2 the nonlinear index of refraction caused

by nonlinearity χ(3), , , n4 is the quadratic nonlinearity index

caused by nonlinearity χ(5), , β3 is third�order dispersion (TOD), , τR isthe time�constant determined by the slope of the curve of Raman amplification in the vicinity of carrier

frequency ω0, , , , β4 is the fourth order disper�

sion, , and β5 the fifth order dispersion. In the wavelength range 1.5–2 μm the

effective mode area equals Aeff = 50–80 μm2 [1]. Factors α5 and α7 in (1) take into account the nonlinearinertial response of the medium caused by Raman scattering in the fiber; they are determined by the quan�

( )

2 3 4 52 4

1 2 3 4 8 92 3 4 5

2 2 46 5 7 ,

E E E E Ei E E E E i i

i E E i E E i E E

∂ ∂ ∂ ∂ ∂+ α + α +α = α + α + α

∂ξ ∂τ ∂τ ∂τ ∂τ

∂ ∂ ∂+ α + α + α

∂τ ∂τ ∂τ

( ) ( ) 1 20, ,E A Pξ τ = ξ τ

( )2

2 0zξ = β τ

02 ln 2τ ( ) 0gt z vτ = − τ 1 2 212

α = − β β

2 22 0 0 2N Pα = = γ τ β ( )effc2 0n Aγ = ω

2 2 23 1 1 0 0 2N Pα = = γ τ β ( ) eff1 4 2n n Aγ = γ

( )4 3 2 06α = β β τ ( )25 0RNα = τ τ

( )26 0 02Nα = − ω τ ( )2

7 1 0RNα = τ τ ( ) ( )28 4 2 024α = − β β τ

( ) ( )39 5 2 0120α = − β β τ

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

COMPUTER SIMULATION OF THE PROPAGATION 239

tities χ(3) and χ(5), respectively. The factor α6 describes self�steepening of laser pulse, while α1, α4, α8 and α9

take into account the second�, third�, fourth�, and fifth�order dispersion, respectively. Coefficients α2 andα3 stand for the self�phase modulation and are determined by, respectively, nonlinearities χ(3) and χ(5).

The shifted dispersion characteristics of single mode fibers considered here are determined by expres�sions

(ps)2/km, (2)

(ps)3/km, (3)

where the values of coefficients are determined experimentally [1].

In order to qualitatively understand the evolution of the time�profile and spectrum of a pulse travellingin a single�mode fiber, we should know the characteristic distances where both linear dispersion terms ofall orders and nonlinear terms begin to be displayed. The characteristic distances, which correspond todispersion spreading of the pulse and are determined by the dispersion terms of the second, third, fourth,and fifth orders, are defined, respectively, as

(4)

The distances, which correspond to self�phase modulation, front steepening, and Raman scattering, aredefined as, respectively,

(5)

It is obvious that the evolution of pulse is determined by the ratio of the lengths above to the fiber length. Wenow give estimations of characteristic lengths for the case of extremely short femtosecond pulse with the cen�tral wavelength λ0 equal to 1.595 μm and durations of both 4T0 and 6T0 (with T0 = λ0/c being the period ofoscillations). This wavelength corresponds to the zero dispersion in a single�mode fiber with a shifted disper�sion characteristic. For this type of fiber, the time of the Raman response is, in the above�indicated spectralrange, τR = 3 fs and we choose the effective area of the mode to be Aeff = 50 μm2 which corresponds to thefollowing values of nonlinearity factors: γ = 3 W–1 km–1 and γ1 = 8.437 × 10–7 W–2 km–1. These parametersof the pulse and the medium give the values of dispersion parameters of second, third, fourth, and fifthorders equal to, respectively, β2 = –0.015 ps2/km, β3 = –0.01 ps3/km, β4 = 1.429 × 10�4 ps4/km, β5 =

⎯1.102 × 10–6 ps5/km. In particular, for a pulse with 4T0 = 21 fs and the peak power =113 kW (N2 = 10), the characteristic distances equal

For a pulse with 6T0 = 32 fs and P0 = 50.2 kW (N2 = 10) these values are

The above�listed values of characteristic distances show that in case of a pulse with 4T0 = 21 fs and the fiberlength ZDisp2, the fifth�order dispersion term ZDisp5 and the Raman scattering caused by the fifth�order non�linearity do not affect the evolution of the pulse. Thus, if the fiber length equals the dispersion length ZDisp2,the linear and nonlinear fifth�order terms in HONSE may practically be neglected. As to the pulse with6T0 = 32 fs, if the fiber length is again ZDisp2 then for the numerical simulation of pulse propagation in asingle�mode fiber with shifted dispersion, the linear dispersion terms and nonlinear terms should be takeninto account only up to third order.

Consider now the numerical solution of HONSE by the method of lines. We have earlier obtained sucha solution of NSE with allowance for terms up to third order [17] in order to analyze the propagation ofoptical soliton in a single�mode fiber.

( )2 3

2 392.3 1140.2 1013.8 284.1β λ = − + λ − λ + λ

( ) ( )5 2 23 5.3052 10 1140.2 2027.6 852.3−

β λ = − × λ − λ + λ

2 3 4 52 0 2 3 0 3 4 0 4 5 0 52 , 6 , 24 , 120 .Disp Disp Disp DispZ Z Z Z= τ β = τ β = τ β = τ β

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

23 0 5 1 0 0 0 0 3 0 0

25 0 1 0

1 , 1 , 2 , ,

.

SPM SPM SS IRS R

IRS R

Z P Z P Z P Z P

Z P

= γ = γ = ω τ γ = τ τ γ

= τ τ γ

( )2

0 2 010P = β λ γτ

2 3 2 4 2 5 2

3 2 5 2 2

3 2 5 2

59 , 5.743 0.1 , 34 0.576 , 474 8 ,

2.951 0.05 , 93 1.576 , 37 0.627 ,

21 0.356 , 658 11.15 .

Disp Disp Disp Disp Disp Disp Disp

SPM Disp SPM Disp SS Disp

IRS Disp IRS Disp

Z m Z m Z Z m Z Z m Z

Z m Z Z m Z Z m Z

Z m Z Z m Z

= = ≈ = ≈ = ≈

= ≈ = ≈ = ≈

= ≈ = ≈

2 3 2 4 2 5 2

3 2 5 2 2

3 2 5 2

133 , 19 0.143 , 174 1.31 , 3599 27 ,

6.639 0.05 , 470 3.534 , 125 0.94 ,

71 0.534 , 4999 37.58 .

Disp Disp Disp Disp Disp Disp Disp

SPM Disp SPM Disp SS Disp

IRS Disp IRS Disp

Z m Z m Z Z m Z Z m Z

Z m Z Z m Z Z m Z

Z m Z Z m Z

= = ≈ = ≈ = ≈

= ≈ = ≈ = ≈

= ≈ = ≈

240

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

HOVHANNISYAN et al.

3. REDUCTION OF HONSE TO THE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS AND NUMERICAL SOLUTION

Partial differential equation (1) is the first�order equation with respect to ξ and fifth�order with respectto τ. The initial condition is chosen to be

(6)

where ω0 = 2πc/λ0. We write E(ξ, τ) in the form E(ξ, τ) = v(ξ,τ) + iw(ξ,τ), with v(ξ,τ) and w(ξ,τ) beingthe real and imaginary parts of the electric field of the pulse. For the complete formulation of problem,also two boundary conditions (BC) must be considered. However, if equation (1) is solved on a sufficientlylong time domain, but variations of the solution are considered on a sufficiently short time interval, theeffect of BC may be neglected. We consider the solution to the equation (1) inside the rectangle 0 ≤ ξ ≤ L,0 ≤ τ ≤ T, divided by straight lines τm = mh (m = 0,1,2, … M) and ξn = nk (n = 0,1,2, … N), where h =T/M, k = L/N. Replacing in equation (1) the partial derivatives with respect to time by the known finite�difference approximations [14,15], we obtain a system of ordinary differential equations with respect tothe variable ξ:

(7)

where

(8)

is the fourth�order finite�difference approximation to the first derivative ,

(9)

( ) { }2

0 020

0, exp exp ,2tE E i

⎛ ⎞ξ = τ = − − ω τ⎜ ⎟

τ⎝ ⎠

( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

21 2 3 4

25 6 6 7

8 9

2 2 2 2

2 2 2 2 2 2 2 2

,

v mw m w m w m w m w m m

w m m m w m m w m m w m

w

ττ τττ

τ τ τ

τ

ττττ τττττ

∂= −α ττ − α − α + α

∂ξ

⎡ ⎤+α +α + α + α ⎣ ⎦+α + α

v v v

v v v v v v v v

v

( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

21 2 3 4

25 6 6 7

8 9

2 2 2 2

2 2 2 2 2 2 2 2

,

w mm w m m w m m w m

w m w m w m w m w m w m w m w m

w

ττ τττ

τ τ τ

τ

ττττ

∂= α ττ + α + α + α

∂ξ

⎡ ⎤+α + α + α + α ⎣ ⎦−α + α

v v v v v

v v v v

v

( )

( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ]

( )( ) ( ) ( ) ( )

( )

( )( ) ( ) ( ) ( )

( )

1 12 25 1 48 2 36 3 16 4 3 5 ; 1

1 12 3 1 10 2 18 3 6 4 5 ; 2

1 12 2 8 1 8 1 2 ; 3,4,... 2

4 6 3 18 2 10 11 12 ; 1

3

3 4 16 3 36 2 48 11 12 ;

25

dx m

dx v m

dx m m m m m Mm M M M M

dx m MM

M M M Mdx m M

M

τ

⎧

− + − + − =

− − + − + =

− − − + + − + = −= ⎨ − − + − − − + −⎡ ⎤

= −⎢ ⎥+⎣ ⎦

− − − + − − −⎡ ⎤=⎢ ⎥+⎣ ⎦

v v v v v

v v v v

v v v v

vv v v v

v

v v v v

v

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎩

∂ ∂τv

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( )( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( )( ) ( ) ( ) ( )

( ) ( )

( )

2

2

2

2

2

1 12 45 1 154 2 214 3 156 4 61 5 10 6 ; 1

45 154 1 214 2 156 31 12 ;

61 4 10 5

1 12 10 1 15 2 4 3 14 4 6 5 6 ; 2

10 15 1 4 2 14 31 12 ; 1

6 4 5

1 12

dx m

M M M Mdx m M

M M

m dx m

M M M Mdx m M

M M

dx

ττ

− + − + − =

− − + − − −⎡ ⎤=⎢ ⎥+ − − −⎣ ⎦

= − − + − + =

− − − − + −⎡ ⎤= −⎢ ⎥− − + −⎣ ⎦

−

v v v v v v

v v v v

v v

v v v v v v v

v v v v

v v

( ) ( ) ( )

( ) ( )

2 16 1 30; 3,4,... 2

16 1 2

m m mm M

m m

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪

− + − −⎡ ⎤⎪ = −⎢ ⎥⎪ + + − +⎩ ⎣ ⎦

v v v

v v

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

COMPUTER SIMULATION OF THE PROPAGATION 241

is the fourth�order approximation to the second�order derivative with the Dirichlet conditions at the leftand right points of the boundary,

(10)

is the central seven�point finite�difference approximation to the third�order derivative, and vττττ

(m),vτττττ

(m) are the fourth�order approximations to, respectively, the functions vττ

(m) and vτττ

(m) with theDirichlet conditions at the left and right points of the boundary. At the ends of τ�intervals (m = 1, 2, 3, M�2,M�1, M), the values of derivatives v

τττ, v

ττττ and v

τττττ are taken to be zero, because these boundaries areassumed to not affect the solution. The derivatives of all orders of w are calculated in the same way as of

v, from formulas (7)–(10). In Eqs. (7), the quantity is the value of the pulseenvelope at the discrete time moment m, is the first derivative of the pulse envelope, which iscalculated by the finite�difference scheme (8). We solve the system (7) by the Runge–Kutta method [16].The relative and absolute errors were in calculations chosen to be 10–6.

4. WIGNER FUNCTION ANALYSIS OF SPECTRAL SUPERCONTINUUM GENERATED DURING PROPAGATION

Traditional methods of studies of supercontinuum, based on the Fourier transformation, do not allowsufficiently exact revealing the presence and position of local peculiarities in the time�profile of anextremely short fs laser pulse, because the basis functions are unbounded and the time�frequency resolu�tion is insufficient. The first shortcoming may be removed by employing the wavelet analysis, although inthis case some uncertainty is retained, since the result depends on the used specific basis function – thewavelet. For overcoming the difficulties associated with the second shortcoming, it seems to be reasonableto employ the Wigner transformation (WT), which has a good resolution in the time�frequency plane, thusit will enable us to reveal effectively the features of the time�frequency structure of the femtosecond pulse[13]. In this Section we perform the WT�based time�frequency analysis of fs�pulse propagation in a single�mode fiber. The time�profile of the pulse E(ξ,τ) is found from the numerical solution of HONSE by themethod of lines. For a fixed value ξ of the fiber length the Wigner function may be represented as [13]

(11)

Computations of both HONSE solution and WT were performed by means of Matlab 9.X. software. Fig�ures 1a–5a show 2D Wigner functions for a pulse of duration 4T0 and peak power 113 kW for the propa�gation distances ξ = 0; 0.1; 0.4; 0.6 and 1, respectively. For the same distances, Figs. 1b–5b illustrate the

time�profiles of pulse intensity obtained by frequency integration of Exp.(11),

(12)

The spectral density of the pulse,

(13)

is for the same distances depicted in Figs. 1c–5c. According to estimations of characteristic distances(Section 2), at ξ = 0.1 the temporal and spectral evolution of the pulse is determined by combined actionof the linear third�order dispersion (ZDisp3 = 0.1ZDisp2) and self�phase modulation. The latter is determinedby the third order nonlinearity χ(3) (ZSPM3 = 0.05ZDisp2). The relative width of the pulse spectrum, Δω/ω0,equals 0.063 for ξ = 0.1. For ξ = 0.4, in addition to the above�indicated mechanisms, the pulse evolutionis affected by also the Raman scattering (ZIRS3 = 0.356 ZDisp2) caused by χ(3). Therefore, Δω/ω0=0.142 for

( )

( )( ) ( ) ( ) ( )

( ) ( )3

0, 1,2,3

0, 2, 1,

3 8 2 13 1 13 11 8 ; 4,5,... 3

8 2 3

m

m m M M M

m m m mdx m M

m m

τττ

⎧⎪ =⎪⎪

= = − −⎨⎪ − − − + − − +⎡ ⎤⎪ = −⎢ ⎥+ + − +⎪⎩ ⎣ ⎦

v

v v v v

v v

( ) ( ) ( )2 22 2w m m w m= +v v

( )2 2w mτ

v

( ) ( ) ( ) ( )*1 1

1 1, , , exp .2 2

W E E j d

∞

ξ

−∞

τ ττ ω = ξ τ+ ξ τ− − ωτ τ∫

( )2,E ξ τ

( ) ( )2, , .E W d

∞

ξ

−∞

ξ τ = τ ω ω∫

( ) ( )2, , ,E W d

∞

ξ

−∞

ξ ω = τ ω τ∫

242

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

HOVHANNISYAN et al.

–0.05

0

0.05

1008060

Wigner distribution – W (t, ω)

(ω – ω0)/ω0

(a)

(b) (c)

0.75

0.50

0.25

200–20 40–40

1.00

0

IE (

ξ =

0,

t) I

IE (

ξ =

0,

ω)I

0.75

0.50

0.25

0.050–0.05 0.10–0.10

1.00

ω0t

ω0 t ( ω – ω0)/ω0

Fig. 1. (a) 2D Wigner distribution; (b) time�profile of the pulse intensity; (c) spectral density of the pulse; ξ = 0.

–400

–20 10 20 40

0.25

0.50

0.75

1.00

IE(ξ

= 0

.1,

t)I

(ω – ω0)/ω0

ω0t–0.100

–0.05 0 0.05 0.10

0.25

0.50

0.75

1.00

IE(ξ

= 0

.1,

ω)I

(ω – ω0)/ω0

400.10

60 80 100 120

0.05

0

–0.05

–0.10

ω0t

Wigner distribution – W(t, ω)

(a)

(b) (c)

Fig. 2. (a) 2D Wigner distribution; (b) time�profile of the pulse intensity; (c) spectral density of the pulse; ξ = 0.1.

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

COMPUTER SIMULATION OF THE PROPAGATION 243

–500

–25 10 25 50

0.25

0.50

0.75

1.00

IE(ξ

= 0

.4,

t)I

(ω – ω0)/ω0

ω0t–0.20

–0.1 0 0.1 0.2

0.25

0.50

0.75

1.00

IE(ξ

= 0

.4, ω

)I

(ω – ω0)/ω0

50 100 150

0.05

0

–0.05

ω0t

Wigner distribution – W(t, ω)(a)

(b) (c)

Fig. 3. (a) 2D Wigner distribution; (b) time�profile of the pulse intensity; (c) spectral density of the pulse; ξ = 0.4.

–800

–40 0 20 40

0.25

0.50

0.75

1.00

IE(ξ

= 0

.6,

t)I

(ω – ω0)/ω0

ω0t–0.20

–0.1 0 0.1 0.2

0.25

0.50

0.75

1.00

IE(ξ

= 0

.6,

ω)I

(ω – ω0)/ω0

00.10

50 100 150

0.05

0

–0.05

–0.10

ω0t

Wigner distribution – W(t, ω)(a)

(b) (c)

–20–60

Fig. 4. (a) 2D Wigner distribution; (b) time�profile of the pulse intensity; (c) spectral density of the pulse; ξ = 0.6.

244

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

HOVHANNISYAN et al.

–1000

–50 0 25 50

0.25

0.50

0.75

1.00

IE(ξ

= 1

, t)

I

(ω – ω0)/ω0

ω0t–0.2

0–0.1 0 0.1 0.2

0.25

0.50

0.75

1.00

IE(ξ

= 1

, ω

)I

(ω – ω0)/ω0

50 100 150

0.05

0

–0.05

ω0t

Wigner distribution – W(t, ω)(a)

(b) (c)

0

–0.10

–0.15

–75 –25

Fig. 5. (a) 2D Wigner distribution; (b) time�profile of the pulse intensity; (c) spectral density of the pulse; ξ = 1.

0.025

0

–0.025

200–20

Δω

(ξ =

0.1

, t)

/ω0

(a) (b)

(c)

0.15

0

–0.15

0–20–40–60

0.3

–0.3

0.2

0

–0.2

–25–50–75 0–100–0.4

0.4

ω0t

ω0t ω0t

(d)

0.050

–0.050

Δω

(ξ

= 0

.4,

t)/ω

0

0.1

0

–0.1

–20–35–50

ω0t

0.2

–0.2–50–65

Δω

(ξ =

0.6

, t)

/ω0

Δω

(ξ =

1,

t)/ω

0

12

Fig. 6. Time dependence of the instantaneous pulse frequency: (a) ξ = 0.1; (b) ξ = 0.4; (c) ξ = 0.6; (d) ξ = 1; the pulsepower⎯113 kW.

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

COMPUTER SIMULATION OF THE PROPAGATION 245

ξ = 0.4. In further propagation, for ξ = 0.6, the linear fourth�order dispersion and self�steepening via χ(3)

(ZDisp4 = 0.576ZDisp2, ZSS = 0.627ZDisp2) are involved and we observe Δω/ω0=0.166. For ξ = 1 this quantityis already 0.211 because the linear second�order dispersion joins in. As seen in Fig.1, the pulse spectrumsuffers, starting from ξ = 0.4, an asymmetric splitting into low� and high�frequency components. This iscaused mainly by the nonlinear inertial Raman response of the medium. From the same distance ξ = 0.4,asymmetry of also the pulse time�profile is displayed (Fig. 3b) and from ξ = 0.6 steepening of the pulseleading front begins.

Figure 6 demonstrates the time dependence of the pulse’ instantaneous frequency at the distances indi�cated in the figure. These plots were computed in time intervals corresponding to the level higher than orequal to 0.1|E(ξ, τ)|max. It is apparent that the linear chirp becomes essentially nonlinear with the increasein the distance travelled. Asymmetry of broadening of the spectrum of extremely short pulse should betaken into account in the problems of compression of femtosecond laser pulses [1].

We also calculated the dependence of the relative width of the pulse spectrum on the travelled distancefor different values of the pulse power. These dependences are shown in Fig. 7 for the indicated values ofpower. The data for these plots were computed in time intervals corresponding to the level higher than orequal to 0.1|E(ξ, τ)|max. This figure demonstrates that the relative spectrum width increases with theincrease in both travelled distance and pulse power. For example, at ξ = 1 and power 113 kW, we observeΔω/ω0 = 0.21.

It is apparent in what has been stated above that the Wigner function really allows revealing the time�frequency dynamics of pulse propagation. In this case, for obtaining more information on the pulse, it isdesirable to analyze also the temporal profile, the spectral density, and the time dependence of the instan�taneous frequency.

The results described here, may be used for elaboration of broadband sources based on exploiting thephenomenon of spectral supercontinuum generation by extremely short femtosecond laser pulses.

0.220

0.205

0.165

0.125

0.085

0.0450.60.50.40.30.20.10 0.7 0.8 0.9 1.0

1

2

3

4

5

t0 = 4T0 = 21 fs

3⎯P0max = 56 kW4⎯P0max = 79 kW

2⎯P0max = 34 kW1⎯P0max = 13 Kw

5⎯P0max = 113 kWΔ

ω/ω

0

ξ

Fig. 7. Plots of relative spectrum width versus the propagation distance; the inset shows the values of the pulse peak powerfor corresponding curves.

246

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS) Vol. 20 No. 4 2011

HOVHANNISYAN et al.

5. CONCLUSIONS

We have demonstrated the results of theoretical study of propagation of extremely short femtosecondpulse (21 fs) whose central wavelength (1.595 μm) corresponds to zero dispersion, in a single�mode fiberwith shifted dispersion characteristics. We considered HONSE, where fifth�order nonlinearity andfourth� and fifth�order dispersion terms are taken into account, and integrated it numerically by themethod of lines. The time dependence of the instantaneous pulse frequency and the evolution of the pulsespectrum were calculated. The distance dependence of the pulse spectrum width is obtained for differentvalues of pulse power. In order to visualize the time�frequency dynamics of pulse propagation, the Wignerfunction analysis has been used.

Work was in part supported by Ministry of Education and Science of Armenia (MESA), grant no. 11�1c124of State Science Committee of MESA, IRMAS International Associated Laboratory, and the VolkswagenStiftung.

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