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Comparison Between the Charactersof the Influence of NonlinearDispersion and Intrapulse RamanScattering on Stationary WavesIvan M. Uzunov aa Institute of Electronics , Bulgarian Academy of Sciences ,72 Boulevard Lenin, Sofia, 1784, BulgariaPublished online: 01 Mar 2007.

To cite this article: Ivan M. Uzunov (1991) Comparison Between the Characters of theInfluence of Nonlinear Dispersion and Intrapulse Raman Scattering on Stationary Waves,Journal of Modern Optics, 38:10, 1911-1918, DOI: 10.1080/09500349114552021

To link to this article: http://dx.doi.org/10.1080/09500349114552021

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JOURNAL OF MODERN OPTICS, 1991, VOL . 38, NO. 10, 1911-1918

Comparison between the characters of the influence ofnonlinear dispersion and intrapulse Raman scattering onstationary waves

IVAN M . UZUNOV

Institute of Electronics, Bulgarian Academy of Sciences,72 Boulevard Lenin, Sofia 1784, Bulgaria

(Received 5 October 1990 ; revision received 23 November 1990)

Abstract. The action of the nonlinear dispersion and intrapulse Ramanscattering on the stationary waves is investigated . The existence of the stationaryperiodical waves is predicted in the case of the presence of nonlinear dispersion .The action of intrapulse Raman scattering is quite different ; it operates as anonlinear force of dissipation which destroys stationary waves .

1 . IntroductionIn describing the propagation picosecond optical pulses, the nonlinear Schrod-

inger equation (NSE) is known to have stationary solutions of both solitary(solitons) and periodic (knoidal) waves [1] . The complementary physical factorsintrapulse Raman scattering (IRS) [2-7], nonlinear dispersion (ND) [8] and third-order dispersion [5, 9] are believed to determine the behaviour of the femtosecondoptical pulses . The question naturally arises about the action of the femtosecondfactors on the stationary solutions of the NSE . Soliton solutions were reported toexist in the presence of the ND [8] . The IRS is found to shift the carrier frequencyof the fundamental soliton [2-6] . The enumerated results are relative to theinfluence of the ND and IRS on solitons only . It is important to make a moregeneral (including both solitons and stationary periodic waves) considerationallowing simultaneous comparison of the action of the distinct femtosecond factors .Analysis of the phase portrait of the stationary waves offers such possibilities .

The purpose of this work is to investigate the effect of the ND and IRS on thestationary waves by their phase portrait .

2 . Basic equationsThe normalized slowly varying amplitude of the electric field ' of the femto-

second optical pulses satisfies equation [6] :

I a + 2T2~+#101 20+iYI ai( I ,I2~)-iY2 a

3+Y30

eraI2=0,

(1)

where L=E/Eo, C-z/zd, T=(t-z/v)/to, YI =µIN, Y3 = a2ig, k'=(ak/a(0)jwo=v -1 ,k" = (a2k/a(0e)I Coo , k"= (a3k/a(03 )Iw0, k3=k"+ 3k"/vk #=zd/znl, zd - t2 /I k" 1, zn]0=no(n2kIo) -' , Y2=k3(6k"to) -1 , µ1 =T(ttto )-1 and µ2 =T1/to . C and T are thenormalized distance and time in a travelling-coordinate system ; co o (where T= 2n/coo) and k are the carrier frequency and propagation constant ; Eo and t o are the

0950-0340/91 $3 .00 © 1991 Taylor & Francis Ltd

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I. M. Uzunov

initial amplitude and physical duration of the pulse ; n o and n2 are the linearrefractive index and the Kerr coefficient ; Zd and z n , are dimensionless parameterscharacterizing dispersion and nonlinear lengths ; 72 is a parameter characterizing thecontribution of the third-order dispersion ; µl and µ2 are parameters characterizingND and IRS ; T1 is the phenomenological time delay of nonlinearity . The boundsobtained in [10] show that the most essential femtosecond factor is IRS . It isassumed that Y2 = 0 and f =1 in this work . We shall seek the stationary solutions ofequation (1) in the form

0=P(u) exp [ 1 4)(', u)],

(2)where the functions p and 0 are real ; u=i-MC ; the constant M correspond to theinverse soliton velocity shift. The c-dependence of is restricted by the conditions :

4~ = k = constant,(3)

4„ is independent of ~ .

Inserting equation (2) into equation (1) and taking into account equation (3) weobtain

p 2 -p2M+ y,pa =constant .

Having taken the integration constant in equation (4) equal to zero, we obtain forthe following :

=M-iY1P2 .

Using equation (5) the following equation is derived :

P+2Y3P 2P+a lp+oc2p 3 +a 3p5 =0,

(6)

where a l =M'-2k, a2 =2(1-y 1M) and a3=4y1 and the dots over the functions pand 4) denote the derivatives with respect to u . It is to be noted that, when y 3 =0,equations (5) and (6) coincide with those derived in [8] .

3 . Phase portrait of the stationary waves of equation (1) for 71=Y2=Y3=0In such conditions the following Hamiltonian system corresponds to equation

(6) :

8HP=ap =P,

Let us write (7) in the form

a1 p2

4W(P)=- 2 + p2 .

(4)

(5)

OH(7)

The Hamiltonian H is defined by

H=22 +W(p), (8)

where the potential energy W(p) is given by

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where

Q(p, p)= -a lp-2p3 ,

P(P, p) =p

The character of the singular points of equation (9) is determined by the magni-tudes [11]

N1= (QP -PP )2 + 4QPPP ,

N2 = QPPP - QPPP ,N3=Q.+ PP ,

where the lower indices p and p denote the corresponding derivatives . The singularpoints are

M1 : P1=P1=0,al l~z

M2,3 : P2,3=± -a

,

p2,3=0 .( 2

The values of Nj at Mj, j =1- 3, are presented in table 1 . Nj and Mj in the specialcase a1 =-1, a2 =2, are also shown in this table . We see that M1 is a saddle point(N'1, N'2>0, N3=0) and M'2 and M3 are centres . The corresponding energy andphase portrait of the stationary waves are presented in figures 1 (a) and (b)respectively. The separatrix curve 2 and curve I near the separatrix are known tocorrespond to the soliton solutions and the stationary periodic (knoidal) waves ofequation (1), respectively (e.g. [12]) .

4. Phase portrait of the stationary waves of equation (1) for yl 00, y 2 =y3 =0In such conditions, the Hamiltonian system corresponding to equation (1) may

be written in the form (7) . The Hamiltonian H is (8) but the potential energy W isdefined by :

M1

Influences on stationary waves

1913

Table 1

M2,3

M'1

M' 2,3

N1

-4a 1

8a 1

N1

4

-8N2

-a l

2a1

N'2

1

-2N3

0

0

N3

0

0

dp Q(P,p)dp P(P, p)

(9)

2a l p a2p4 cc o

(10)W(P) = 2 + 4 + 6

The system (7) can be written in the form (9), where

Q(P,p)= -a lp-a2p3 -a3p5 ,

P(P, p) =p

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',4

1,2

0 .8

0,

0.4

0,

0

-0,2

0,8-

0.6-

0 .4-

0.2-

0

-0,2-

-0,4-

-0,6-

-0,8-

-1,2

-J8 -0,4

Impr

I. M. Uzunov

0

0,4

0,8

(a)

PA

1,2 1,6

I

,

I

I

-0,8

-0,4

0

0,4

0,8

1,2

(b)

s

Figure 1 . (a) Potential energy of equation (7) when y1=Y2=y3=0 (a l =-1; a 2 =2). (b)Phase portrait of the stationary waves of equation (1) when y1=Y2=y3 =0 (a l =-1 ;a2 = 2) .

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az

Ca2

al\1nNZ -al 4a, --+az -4-a3

a3

0"

N3 0Y3 Ca3-Ca -4

a3\1/2,

Influences on stationary waves

1915

The singular points are

R1 : pi =p1=0,

IXz

az z

al 1/2Rz,3 : Pz,3=±

G--+ C- -4-

'2) 1/2 , pz,3=0 .

(11)IX3

a3

a 3

The values Nj at Rj , j =1-3 , are shown in table 2, where Y3=0' Nj and Rj, j =1-3 ,in the special case a l = - 1, a2=2, a 3 =1, are also presented in this table . Similar tothe above paragraph about M;, R1 is a saddle point (Ni, N2>0; N3=0), and RZand R3 are centres . The potential energy and the phase portrait of the stationarywaves are shown in figures 2 (a) and (b) respectively. The separatrix curve 2 isbelieved to correspond to soliton solution of equation (1) (y 1 :A 0 ; Y z = y 3 =0)obtained in [8] . The question naturally arises about the existence of the stationaryperiodic solutions of the same equation . As far as we know, such a matter has notbeen considered yet . Having started from the well known correspondence betweenthe phase portrait of equation (1) (y 1 =Yz = Y3 = 0) and its stationary solutions,taking into account the complete topological equivalence between the phaseportraits in figure 1 (b) and figure 2 (b) and the existence of the soliton solution ofequation (1) (Y1 :A 0 ; yz = y 3 =0), we reach the conclusion that the latter equation hasstationary periodic waves .

Here the choice of the magnitude of the parameter a 3 = I must be explained, as itis known that a 3 ,: 10-4-10-3 in practical situations [10] . Such a value of a 3 hasbeen chosen in order to show that topological equivalence between the phaseportraits in figure 1 (b) and figure 2 (b) is preserved even in this case .

To sum up this paragraph, it may be said that in the presence of the ND, theexistence of not only solitons but also stationary periodic waves is possible .

5 . The action of the IRS on the phase portrait of the stationary waves(equation (1) when yl, y3 :A 0 ; Y2 =0)When IRS is taken into account, the following non-Hamiltonian system

corresponds to equation (6) :

Table 2 .

Ri

Rs,3

Ri

Rz,3

N1 -4a, 16a,-4 a3 +2Y3 \a3/z-4v3 \a3/

AP,

4

-9.37+0 68Y3

//az

aZ

al liz+(4az -2y 3 -~

a1L'3)OC3

'3)

NZ

1

-2.32

N'3

0

-0.83Y3

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I. M. Uzunov

W 1

3,5

3

2,5 -

1,5

0,5-

0

-0.5 1'6

-1,2

-0,8

-0.4 T 0

0,4

0,8

1,2

1,6

(a)

P

0,8

0,6

0,4

0,

0

-0,2 -

-0,4

-0,6

-12 -0,8

-0,4

(b)

0,4 .8

1,2

Figure 2 . (a) Potential energy of equation (7) when y 1 :AO; Y 2 =y 3 =0 (a 1 =-1 ; a 2 =2 ;a3 =1). (b) Phase portrait of the stationary waves of equation (1) when y l :?-, 0 ;Y2=Y3=0 (a 1 =-1 ; a2 =2; a 3 =1) .

3

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Influences on stationary waves

1917

OHP =ap =P,

aH+Q dfi_ - ap

,

( 12)

where H and W(p) are defined by equations (8) and (10) . As we see from equation(12), the term Qd = -2y 3pp2 is the nonlinear force of dissipation . The systems (12)can be written in the form (9) where

Q(P, P) = - a1p - a2p3- oc3p5- 2y3pp 2 ,P(P, p) =p •

The singular points of equation (9) coincide with those of equation (11) . Themagnitudes ofNj at Rj ,j = 1-3, are presented in table 2, where y 3 :0 . Nj and Rj inthe special case a = -1, a 2 = 2, a3 =1, are also shown in this table . Depending onthe magnitude of y3 , two different cases are generated : firstly, 0 < y 3 < 3 .7 ; secondly,y 3 > 3 .7 . For typical parameters of the femtosecond pulses in optical fibres,y 3: 10-3 [10] . In this case, R1 remains a saddle point but RZ and R3 convert intofocuses . This result is illustrated in figure 3 . As we can see, the influence of the IRSon the stationary waves is quite different from that of the ND . Acting as a nonlinearforce of dissipation, the IRS destroys the stationary periodic and stationary solitary(solitons) waves . The last phase portrait is built by means of the isocline method .The large magnitude of y 3 (y3 = 1) is assumed to show the influence of the nonlineardissipation clearly . In the second case (y 3 > 3 . 7), R1 retains saddle-point but R2 andR3 turn into knots. It is to be noted that the change in the character of the singular

1

0 .5

-05

. 1,5•

Figure 3 . Phase portrait of the stationary waves of equation (1) when y3 #0, y 1 =y 2 =0(a3 =0) .

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Influences on stationary waves

points (from a centre through a focus to a knot) with increase in the nonlineardissipation value y3 is known in the theory of nonlinear waves (for example [12]) .

6. ConclusionAn analysis of the influence of ND and IRS on the stationary waves of the NSE

is performed using a phase portrait . The possibility for the existence of thestationary periodic waves is predicted in the presence of ND . It is shown that,operating as a nonlinear force of dissipation, IRS destroys stationary waves .

In the present author's opinion, the results obtained in this work, on the onehand, proved to be helpful in revealing the character of the effect of ND and IRS onstationary waves and, on the other hand, are a good beginning in the analysis of thepropagation of femtosecond pulses using a phase portrait .

AcknowledgmentsIn conclusion, I would like to thank Professor A . Spasov and Dr L. Pavlov for

their support of this work .

References[1] DAVYDOV, A. S., 1988, Solitons In Molecular Systems (Kiev : Naukova Dumka) .[2] DIANOV, E. M ., KARASIK,A. YA., MANYSHEV, P. V ., PROKHOROV, A. M., SERKIN, V. N .,

STELMAKH, M . F., and FoMICHEV, A. A., 1985, Pis'ma Zh . eksp . teor. Fiz ., 41, 242 .[3] MITSCHE, F . M., and MOLLENAUER, L. F., 1986, Optics Lett ., 11, 659 .[4] GORDON, J . P ., 1986, Optics Lett., 11, 662 .[5] KoDAMA, Y., and HASEGAWA, A., 1987, IEEE Jl quant . Electron ., QE-23, 510 .[6] AKHMANOV, S. A ., VYSLOUKH, V. A ., and CHIRKIN, A. S ., 1988, Optics of Femtosecond

Laser Pulses (Moscow: Nauka) .[7] UZUNOV, I . M ., 1990, Opt. quant . Electron ., 22, 529 .[8] ANDERSON, D ., and LISAK, M., 1983, Phys. Rev . A, 27, 1393 .[9] UzuNOV, I . M., and MITEV, V. M ., 1990, Bulg . J . Phys ., 17, 249 .

[10] POTASEK, M. J ., 1989, J . appl. Phys ., 65, 941 .[11] BOGOLJUBOV, N . N., and MITROPOLSKI, U. A ., 1963, Asymptotic Methods in Theory of

Nonlinear Oscillations (Moscow : Nauka) .[12] RABINOVICH, M. I ., and TRUBETSKOV, D. I ., 1984, Introduction to Theory of Oscillations

and Waves (Moscow: Nauka) .

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