Computer simulation of the propagation of extremely short ...

ISSN 1060992X, Optical Memory and Neural Networks (Information Optics), 2011, Vol. 20, No. 4, pp. 237–246. © Allerton Press, Inc., 2011.

Computer Simulation of the Propagation of Extremely Short Femtosecond Pulse in SingleMode Fiber with Shifted Dispersion Characteristics D. L. Hovhannisyana, V. O. Chaltykyanb, A. O. Hovhannisyana, G. D. Hovhannisyana, and K. A. Hovhannisyana a

Yerevan State University, Armenia for Physical Research, NAS of Armenia email: [email protected]

bInstitute

Received in final form, May 3, 2011

Abstract—We present the results of theoretical study of generation of spectral supercontinuum by an extremely short femtosecond laser pulse with the central wavelength corresponding to zero dispersion, propagating in a singlemode fiber with shifted dispersion characteristics. Highorder nonlinear Schrödinger equation (HONSE) is considered with higherorder 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 and the medium length dependence of the spectrum width is obtained. For study of the timefrequency dynamics of extremely short femtosecond laser pulse propagating in singlemode optical fiber with allowance 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

1. INTRODUCTION For generation of spectral continuum by a laser pulse propagating in a nonlinear medium, the basic point is the presence of laser intensitydependent term in the index of refraction of the medium with Kerr nonlinearity. Appearance of the intensity dependent term leads in case of short laser pulses to physically meaningful modulation of the laser pulse phase, i.e., selfphases modulation (SPM). Within the frames of the approximation taking into account only the first order of material dispersion the temporal selfaction of laser pulse leads to symmetric broadening of its spectrum. However, a number of physical mechanisms lead to asymmetry of spectral broadening even at moderate intensities of laser pulse. Three most important mechanisms are associated with spatial selfaction, formation of shockwave front of pulse envelope, and finite time of nonlinear response of the medium. Spatial selfaction is asso ciated with the Kerr nonlinearity of the medium [1]. The formation of the shock wave is caused by the intensity dependence of the pulse group velocity [1]. Effects connected with the finite time of nonlinear response of the medium become especially considerable for fewcycle 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 delaying nonlinearity suffers lowfrequency shift. Spectral broadening induced by the delayed nonlinearity is thus equivalent to stimulated Raman scattering (SRS). At excitation of Raman resonance by a fewcycle laser pulse the field spectral component shifted by the frequency 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 selfaction. At the expense of excita tion of molecular vibrations energy redistribution occurs in the pulse spectrum and this leads to a red shift of the spectrum peak [1]. During propagation of an extremely short laser pulse, fewcycle pulse, in an iso tropic nonlinear medium, if the central wavelength of the pulse is in the range of zero dispersion of the medium, dispersion effects at the early stage of propagation will be determined by higherorder (cubic and higher) terms in the expansion of the index of refraction. However, during further propagation shift of the central wavelength occurs because of spectral broadening caused by both noninertial and inertial nonlin 237

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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 in the 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 to zero dispersion, from 1300 to 800 nm, which coincides with the central wavelength of titaniumsapphire femtosecond laser. The mode dispersion may completely be eliminated, if structural parameters of the fiber will be chosen so that only one, fundamental, mode will travel in the lightguide. Singlemode optical fibers have this property. But the dispersion may also be shifted by means of quartz glass doping. Such lightguides are called optical fibers with shifted dispersion; they can have zero dispersion at the wavelength with minimum damping (1550 nm). In such fibers the second value is observed of the central wavelength corresponding 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 laser pulse, HONSE may be used where both dispersion terms and higher order nonlinearities are taken into account. 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 zerodispersion range, in a singlemode fiber with shifted dis persion characteristics. We consider HONSE where the fifthorder nonlinearity and the fourth and fifth order dispersion terms are taken into account. Numerical integration of such HONSE is performed by the method 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. The timefrequency 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 the zaxis in a singlemode fiber. This process is described by HONSE which can be represented in the form (notations of [1]) 2 3 4 5 2 4 i ∂E + α1 ∂ E2 + α 2 E E +α 3 E E = iα 4 ∂ E3 + α 8 ∂ E4 +iα 9 ∂ E5 ∂ξ ∂τ ∂τ ∂τ ∂τ 2 2 4 + iα 6 ∂ E E + iα 5 E ∂ E + iα 7 E ∂ E , ∂τ ∂τ ∂τ

(

(1)

)

where E ( ξ, τ) = A ( ξ, τ) P01 2 , A(ξ,τ) is the slowly varying amplitude, P0 the peak power of the pulse, ξ = ( z β 2 ) τ 02 , β2 is the group velocity dispersion, τ0 the pulse duration determined at halfmaximum THM ≡ 2 ln 2τ 0 = 1.665τ0 (for a Gaussian pulse), τ = (t − z v g ) τ 0 , vg is the group velocity, α1 = − 1 β 2 β 2 , 2 α 2 = N 2 = γ P0τ 02 β 2 , γ = n2 ω0 ( c Aeff ) 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), α 3 = N 12 = γ 1P02 τ 20 β 2 , γ 1 = γ ( n4 n2 ) Aeff , n4 is the quadratic nonlinearity index caused by nonlinearity χ(5), α 4 = β3 (6 β2 τ 0 ) , β3 is thirdorder dispersion (TOD), α 5 = N 2 ( τ R τ 0 ) , τR is the timeconstant determined by the slope of the curve of Raman amplification in the vicinity of carrier frequency ω0, α 6 = − 2N 2 ( ω0τ 0 ) , α 7 = N 12 ( τ R τ 0 ) , α 8 = −(β 4 24 ) β 2 τ 02 , β4 is the fourth order disper

(

)

(

)

sion, α 9 = −(β5 120) , 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 nonlinear inertial response of the medium caused by Raman scattering in the fiber; they are determined by the quan β 2 τ30

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tities χ(3) and χ(5), respectively. The factor α6 describes selfsteepening of laser pulse, while α1, α4, α8 and α9 take into account the second, third, fourth, and fifthorder dispersion, respectively. Coefficients α2 and α3 stand for the selfphase 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

β 2 ( λ ) = − 392.3 + 1140.2λ − 1013.8λ 2 + 284.1λ 3 (ps)2/km,

(

(2)

)

β3 ( λ ) = − 5.3052 × 10 −5 λ 2 1140.2 − 2027.6λ + 852.3λ 2 (ps)3/km,

(3)

where the values of coefficients are determined experimentally [1]. In order to qualitatively understand the evolution of the timeprofile and spectrum of a pulse travelling in a singlemode fiber, we should know the characteristic distances where both linear dispersion terms of all orders and nonlinear terms begin to be displayed. The characteristic distances, which correspond to dispersion spreading of the pulse and are determined by the dispersion terms of the second, third, fourth, and fifth orders, are defined, respectively, as Z Disp2 =2τ 02 β 2 , Z Disp3 =6τ30 β3 , Z Disp4 =24 τ 04 β 4 , Z Disp5 = 120τ50 β5 .

(4)

The distances, which correspond to selfphase modulation, front steepening, and Raman scattering, are defined as, respectively,

(

)

Z SPM 3 = 1 ( γ P0 ) , Z SPM 5 = 1 γ1 P02 , Z SS = ( ω0 τ0 ) ( 2γP0 ) , Z IRS3 = ( τ0 τ R ) ( γP0 ) ,

(

)

(5)

Z IRS5 = ( τ0 τ R ) γ1P02 .

It is obvious that the evolution of pulse is determined by the ratio of the lengths above to the fiber length. We now 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 of oscillations). This wavelength corresponds to the zero dispersion in a singlemode fiber with a shifted disper sion characteristic. For this type of fiber, the time of the Raman response is, in the aboveindicated spectral range, τR = 3 fs and we choose the effective area of the mode to be Aeff = 50 μm2 which corresponds to the following values of nonlinearity factors: γ = 3 W–1 km–1 and γ1 = 8.437 × 10–7 W–2 km–1. These parameters of the pulse and the medium give the values of dispersion parameters of second, third, fourth, and fifth orders equal to, respectively, β2 = –0.015 ps2/km, β3 = –0.01 ps3/km, β4 = 1.429 × 104 ps4/km, β5 = ⎯1.102 × 10–6 ps5/km. In particular, for a pulse with 4T0 = 21 fs and the peak power P0 = 10 β 2 ( λ ) γτ 02 = 113 kW (N 2 = 10), the characteristic distances equal

Z Disp2 = 59 m, Z Disp3 = 5.743 m ≈ 0.1 Z Disp2, Z Disp4 = 34 m ≈ 0.576 Z Disp2, Z Disp5 = 474 m ≈ 8 Z Disp2, Z SPM 3 = 2.951 m ≈ 0.05 Z Disp2, Z SPM 5 = 93 m ≈ 1.576 Z Disp2, Z SS = 37 m ≈ 0.627 Z Disp2 , Z IRS3 = 21 m ≈ 0.356 Z Disp2 , Z IRS5 = 658 m ≈ 11.15 Z Disp2 . For a pulse with 6T0 = 32 fs and P0 = 50.2 kW (N 2 = 10) these values are

Z Disp2 = 133 m, Z Disp3 = 19 m ≈ 0.143 Z Disp2, Z Disp4 = 174 m ≈ 1.31Z Disp2, Z Disp5 = 3599 m ≈ 27 Z Disp2, Z SPM 3 = 6.639 m ≈ 0.05 Z Disp2, Z SPM 5 = 470 m ≈ 3.534 Z Disp2, Z SS = 125 m ≈ 0.94 Z Disp2 , Z IRS3 = 71 m ≈ 0.534 Z Disp2 , Z IRS5 = 4999 m ≈ 37.58 Z Disp2. The abovelisted values of characteristic distances show that in case of a pulse with 4T0 = 21 fs and the fiber length ZDisp2, the fifthorder dispersion term ZDisp5 and the Raman scattering caused by the fifthorder non linearity do not affect the evolution of the pulse. Thus, if the fiber length equals the dispersion length ZDisp2, the linear and nonlinear fifthorder terms in HONSE may practically be neglected. As to the pulse with 6T0 = 32 fs, if the fiber length is again ZDisp2 then for the numerical simulation of pulse propagation in a singlemode fiber with shifted dispersion, the linear dispersion terms and nonlinear terms should be taken into account only up to third order. Consider now the numerical solution of HONSE by the method of lines. We have earlier obtained such a solution of NSE with allowance for terms up to third order [17] in order to analyze the propagation of optical soliton in a singlemode fiber. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

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3. REDUCTION OF HONSE TO THE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS AND NUMERICAL SOLUTION Partial differential equation (1) is the firstorder equation with respect to ξ and fifthorder with respect to τ. The initial condition is chosen to be ⎛ 2 ⎞ (6) E ( ξ = 0, τ) = E 0 exp ⎜ − t 2 ⎟ exp {−iω0 τ} , ⎝ 2τ 0 ⎠ 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) being the 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 sufficiently long time domain, but variations of the solution are considered on a sufficiently short time interval, the effect 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 to the variable ξ:

∂v ( m) 2 = −α1wττττ ( m) − α 2v2w2 ( m) w ( m) − α3v2w2 ( m) w ( m) + α 4v τττ ( m) ∂ξ 2 + α5v2w2 τ ( m) v ( m) + α 6v τ ( m) v2w2 ( m) + α 6v ( m) v2w2 τ ( m) + α 7v ( m) ⎡⎣v2w2 ( m) ⎤⎦

τ

+ α8wττττ + α 9v τττττ, ∂w ( m) 2 = α1v ττττ ( m) + α 2v2w2 ( m) v ( m) + α3v2w2 ( m) v ( m) + α 4wτττ ( m) ∂ξ 2 + α5v2w2 τ ( m) w ( m) + α6wτ ( m) v2w2 ( m) + α6w ( m) v2w2 τ ( m) + α 7w ( m) ⎡⎣v2w2 ( m) ⎤⎦ τ − α8v ττττ + α9w,

(7)

where

⎧ ⎪ ⎪(1 12dx ) [−25v (1) + 48v ( 2) − 36v (3) + 16v ( 4) − 3v (5)] ; m = 1 ⎪(1 12dx ) [−3v (1) − 10v ( 2) + 18v (3) − 6v ( 4) + v (5)] ; m = 2 ⎪ ⎪⎪(1 12dx ) [v ( m − 2) − 8v ( m − 1) + 8v ( m + 1) − v ( m + 2)] ; m = 3,4,...M − 2 v τ ( m) = ⎨ ⎡−v ( M − 4) + 6v ( M − 3) − 18v ( M − 2) + 10v ( M − 1)⎤ ⎪(1 12dx ) ⎢ ⎥;m = M − 1 ⎣+3v ( M ) ⎦ ⎪ ⎪ ⎡3v ( M − 4) − 16v ( M − 3) + 36v ( M − 2) − 48v ( M − 1)⎤ ⎪(1 12dx ) ⎢ ⎥;m = M ⎪ ⎣+25v ( M ) ⎦ ⎪⎩ is the fourthorder finitedifference approximation to the first derivative ∂v ∂τ ,

⎧ ⎪ 1 12dx 2 ⎪ ⎪ 2 ⎪ 1 12dx ⎪ ⎪ 2 v ττ ( m) = ⎨ 1 12dx ⎪ ⎪ 1 12dx 2 ⎪ ⎪ ⎪ 1 12dx 2 ⎪⎩

( ( ( (

) [45v (1) − 154v (2) + 214v (3) − 156v (4) + 61v (5) − 10v (6)]; m = 1 45v ( M ) − 154v ( M − 1) + 214v ( M − 2) − 156v ( M − 3)⎤ ) ⎡⎢⎣+61v ( M − 4) − 10v ( M − 5) ⎥;m = M ⎦ ) [10v (1) − 15v (2) − 4v (3) + 14v (4) − 6v (5) + v (6)]; m = 2 10v ( M ) − 15v ( M − 1) − 4v ( M − 2) + 14v ( M − 3)⎤ ) ⎡⎢⎣−6v ( M − 4) + v ( M − 5) ⎥;m = M − 1 ⎦

(

) ⎣⎢⎡+16v ( m + 1) − v ( m + 2)

(8)

(9)

−v ( m − 2) + 16v ( m − 1) − 30v ( m)⎤ ⎥ ; m = 3,4,...M − 2 ⎦

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is the fourthorder approximation to the secondorder derivative with the Dirichlet conditions at the left and right points of the boundary,

⎧ ⎪0, m = 1,2,3 ⎪⎪ (10) v τττ ( m) = ⎨0, m = M − 2, M − 1, M ⎪ ⎡v ( m − 3) − 8v ( m − 2) + 13v ( m − 1) − 13v ( m + 1)⎤ ⎪ 1 8dx 3 ⎢ ⎥ ; m = 4,5,...M − 3 ⎪⎩ ⎣+8v ( m + 2) − v ( m + 3) ⎦ is the central sevenpoint finitedifference approximation to the thirdorder derivative, and vττττ(m), vτττττ(m) are the fourthorder approximations to, respectively, the functions vττ(m) and vτττ(m) with the Dirichlet conditions at the left and right points of the boundary. At the ends of τintervals (m = 1, 2, 3, M2, M1, M), the values of derivatives vτττ, vττττ and vτττττ are taken to be zero, because these boundaries are assumed to not affect the solution. The derivatives of all orders of w are calculated in the same way as of 2 2 v, from formulas (7)–(10). In Eqs. (7), the quantity v2w2 ( m) = v ( m) + w ( m) is the value of the pulse envelope at the discrete time moment m, v2w2 τ ( m) is the first derivative of the pulse envelope, which is calculated by the finitedifference 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 allow sufficiently exact revealing the presence and position of local peculiarities in the timeprofile of an extremely short fs laser pulse, because the basis functions are unbounded and the timefrequency resolu tion is insufficient. The first shortcoming may be removed by employing the wavelet analysis, although in this case some uncertainty is retained, since the result depends on the used specific basis function – the wavelet. For overcoming the difficulties associated with the second shortcoming, it seems to be reasonable to employ the Wigner transformation (WT), which has a good resolution in the timefrequency plane, thus it will enable us to reveal effectively the features of the timefrequency structure of the femtosecond pulse [13]. In this Section we perform the WTbased timefrequency analysis of fspulse propagation in a single mode fiber. The timeprofile of the pulse E(ξ,τ) is found from the numerical solution of HONSE by the method of lines. For a fixed value ξ of the fiber length the Wigner function may be represented as [13] ∞

W ξ ( τ,ω ) =

∫ E (ξ,τ+ 2 )E τ1

*

−∞

(ξ,τ−τ2 ) exp(− j ωτ ) d τ .

(11)

1

1

1

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 2 timeprofiles of pulse intensity E ( ξ,τ) obtained by frequency integration of Exp.(11), ∞

∫

E ( ξ,τ ) = Wξ ( τ,ω )d ω. 2

(12)

−∞

The spectral density of the pulse, ∞

∫

E ( ξ,ω ) = Wξ ( τ,ω )d τ, 2

(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 action of the linear thirdorder dispersion (ZDisp3 = 0.1ZDisp2) and selfphase modulation. The latter is determined by 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 aboveindicated mechanisms, the pulse evolution is affected by also the Raman scattering (ZIRS3 = 0.356 ZDisp2) caused by χ(3). Therefore, Δω/ω0=0.142 for OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

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HOVHANNISYAN et al. Wigner distribution – W (t, ω) (a) –0.05 (ω – ω0)/ω0

0 0.05

60

100 1.00

(b)

0.75

IE (ξ = 0, ω)I

IE (ξ = 0, t) I

1.00

80 ω0t

0.50 0.25 0 –40

–20

0 ω0 t

0.75 0.50 0.25 0 –0.1

40

20

(c)

0 0.05 –0.05 ( ω – ω0)/ω0

0.10

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

(a) Wigner distribution – W(t, ω) –0.10 –0.05 (ω – ω0)/ω0

0 0.05 0.10 40

80 ω0t

100

120

(b)

1.00 IE(ξ = 0.1, ω)I

IE(ξ = 0.1, t)I

1.00

60

0.75 0.50 0.25 0 –40

(c)

0.75 0.50 0.25

–20

10 ω0t

20

40

0 –0.10

–0.05

0 0.05 (ω – ω0)/ω0

0.10

Fig. 2. (a) 2D Wigner distribution; (b) timeprofile of the pulse intensity; (c) spectral density of the pulse; ξ = 0.1. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

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(a) Wigner distribution – W(t, ω)

–0.05 (ω – ω0)/ω0

0 0.05

50

150 1.00

(b) IE(ξ = 0.4, ω)I

IE(ξ = 0.4, t)I

1.00

ω0t 100

0.75 0.50 0.25 0 –50

–25

10 ω0t

25

(c)

0.75 0.50 0.25 0 –0.2

50

–0.1

0 0.1 (ω – ω0)/ω0

0.2

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

(a) Wigner distribution – W(t, ω) –0.10 –0.05 (ω – ω0)/ω0

0 0.05 0.10 0

ω0t

100

150

1.00

(b) IE(ξ = 0.6, ω)I

IE(ξ = 0.6, t)I

1.00

50

0.75 0.50 0.25 0 –80 –60 –40 –20 ω0t

0

20

40

(c)

0.75 0.50 0.25 0 –0.2

–0.1

0 0.1 (ω – ω0)/ω0

0.2

Fig. 4. (a) 2D Wigner distribution; (b) timeprofile of the pulse intensity; (c) spectral density of the pulse; ξ = 0.6. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

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HOVHANNISYAN et al. (a) Wigner distribution – W(t, ω) –0.15 –0.10 –0.05

(ω – ω0)/ω0

0 0.05 0

ω0t

100

150

1.00

(b) IE(ξ = 1, ω)I

IE(ξ = 1, t)I

1.00

50

0.75 0.50 0.25 0 –100 –75 –50 –25 ω0t

0

25

(c)

0.75 0.50 0.25 0

50

–0.2

–0.1 0 0.1 (ω – ω0)/ω0

0.2

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

(a)

0.2 Δω (ξ = 0.4, t)/ω0

Δω (ξ = 0.1, t)/ω0

0.050 0.025 0 –0.025 –0.050 –20

(c)

0 –0.15

–40

–20 ω0t

0 –0.1

–50

0.4

0.15

–0.3 –60

0.1

–0.2 –65

20

Δω (ξ = 1, t)/ω0

Δω (ξ = 0.6, t)/ω0

0.3

0 ω0t

(b)

0

–35 ω0t

–20

–50

(d)

0.2 0 –0.2 –0.4 –100

–75

–50 ω0t

–25

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 pulse power⎯113 kW. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

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0.220 t0 = 4T0 = 21 fs 1⎯P0max = 13 Kw 2⎯P0max = 34 kW 3⎯P0max = 56 kW 4⎯P0max = 79 kW 5⎯P0max = 113 kW

0.205

5

0.165

Δω/ω0

4

3

0.125

2 0.085 1

0.045

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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

ξ = 0.4. In further propagation, for ξ = 0.6, the linear fourthorder dispersion and selfsteepening via χ(3) (ZDisp4 = 0.576ZDisp2, ZSS = 0.627ZDisp2) are involved and we observe Δω/ω0=0.166. For ξ = 1 this quantity is already 0.211 because the linear secondorder dispersion joins in. As seen in Fig.1, the pulse spectrum suffers, starting from ξ = 0.4, an asymmetric splitting into low and highfrequency components. This is caused mainly by the nonlinear inertial Raman response of the medium. From the same distance ξ = 0.4, asymmetry of also the pulse timeprofile is displayed (Fig. 3b) and from ξ = 0.6 steepening of the pulse leading 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 or equal to 0.1|E(ξ, τ)|max. It is apparent that the linear chirp becomes essentially nonlinear with the increase in the distance travelled. Asymmetry of broadening of the spectrum of extremely short pulse should be taken 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 distance for different values of the pulse power. These dependences are shown in Fig. 7 for the indicated values of power. The data for these plots were computed in time intervals corresponding to the level higher than or equal to 0.1|E(ξ, τ)|max. This figure demonstrates that the relative spectrum width increases with the increase in both travelled distance and pulse power. For example, at ξ = 1 and power 113 kW, we observe Δω/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 is desirable 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 the phenomenon of spectral supercontinuum generation by extremely short femtosecond laser pulses. OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

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5. CONCLUSIONS We have demonstrated the results of theoretical study of propagation of extremely short femtosecond pulse (21 fs) whose central wavelength (1.595 μm) corresponds to zero dispersion, in a singlemode fiber with shifted dispersion characteristics. We considered HONSE, where fifthorder nonlinearity and fourth and fifthorder dispersion terms are taken into account, and integrated it numerically by the method of lines. The time dependence of the instantaneous pulse frequency and the evolution of the pulse spectrum were calculated. The distance dependence of the pulse spectrum width is obtained for different values of pulse power. In order to visualize the timefrequency dynamics of pulse propagation, the Wigner function analysis has been used. Work was in part supported by Ministry of Education and Science of Armenia (MESA), grant no. 111c124 of State Science Committee of MESA, IRMAS International Associated Laboratory, and the Volkswagen Stiftung. REFERENCES 1. Agrawal, G.P., Nonlinear Fiber Optics, New York: Academic Press, 2001, third ed. 2. Yanovsky, V. and Wise F., Nonlinear Propagation of HighPower, Sub100 fs Pulses Near the ZeroDispersion Wavelength of an Optical Fiber, Optics Letters, 1994, vol. 19, no. 19, pp. 1547–1549. 3. Dumais, P., Gonthier, F., Lacroix, S., Bures, J., Vileneuve, A., Wigley, P.G., and Stegeman, G.I., Enhanced SelfPhase Modulation in Tapered Fibers, Optics Letters, 1993, vol. 18, no. 23, pp. 1993–1995. 4. Gaeta, A.L., Nonlinear Propagation and Continuum Generation in Microstructured Optical Fibers, Optics Letters, 2002, vol. 27, no. 11, pp. 924–926. 5. Kalosha, V.P. and Hermann, J., SelfPhase Modulation and Compression of FewOpticalCycles Pulses, Phys. Rev. A, 2000, vol. 62, p. 011804(R). 6. Karasawa, N., Computer Simulations of Nonlinear Propagation of an Optical Pulse Using a FiniteDifference in the FrequencyDomain Method, IEEE J. Quantum Electron, 2002, vol. 38, pp. 626–629. 7. Blow, K.J. and Wood, D., Theoretical Description of Transient Stimulated Raman Scattering in Optical Fibers, IEEE J. Quantum Electron, 1989, vol. 25, pp. 2665–2673. 8. Brabec, T. and Krausz, F., Nonlinear Optical Pulse Propagation in the SingleCycle Regime, Phys. Rev. Lett., 1997, vol. 78, pp. 3282–3285. 9. Karasawa, N., Nakamura, S., Nakagawa, N., Shibata, M., Morita, R., Shigekawa, H., and Yamashita, M., Comparison Between Theory and Experiment of Nonlinear Propagation for aFewCycle and Ultrabroadband Optical Pulses in a FusedSilica Fiber, IEEE J. Quantum Electron, 2001, vol. 37, pp. 398–404. 10. Kinsler, P., New G.H.C. Wideband Pulse Propagation: SingleField and MultiField Approaches to Raman Interactions, Phys. Rev. A, 2005, vol. 72, p. 033804. 11. Husakou, A.V. and Hermann, J., Supercontinuum Generation of HigherOrder Solitons by Fission in Photonic Crystal Fibers, Phys. Rev. Lett., 2001, vol. 87, p. 203901. 12. Kolesnik, M. and Moloney, J.V., Nonlinear Optical Pulse Propagation Simulation: From Maxwell’s to Unidi rectional Equations, Phys. Rev. E, 2004, vol. 70, p. 036604. 13. Cohen, L., TimeFrequency Distributions—a Review, Proc. IEEE, 1981, vol. 77, pp. 941–981. 14. LeVeque, R.J., Finite Difference Methods for Ordinary and Partial Differential Equations: SteadyState and Time Dependent Problems, Seattle: University of Washington; Washington: Society for Industrial and Applied Math ematics, 2007. 15. Schiesser, W.E. and Griffiths, G.W., A Compendium of Partial Differential Equation Models, Method of Lines Analysis with Matlab, New York: Cambridge University Press, 2009. 16. Mathews, J.H. and Fink, K.D., Numerical Methods Using MATLAB, Upper Saddle River, NJ: Prentice Hall, 1999, p. 07458. 17. Hovhannisyan, D.L., Hovhannisyan, A.H., Hovhannisyan, G.D., and Hovhannisyan, K.A., Numerical Simu lation of Propagation of a Femtosecond Optical Soliton in a SingleMode Fiber With Allowance for the Imag inary Part of Raman Response, Journal of Contemporary Physics (Armenian Academy of Sciences), 2010, vol. 45, no. 6, pp. 251–257.

OPTICAL MEMORY AND NEURAL NETWORKS (INFORMATION OPTICS)

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2011