Dark solitons produced by phase steps in nonlinear optical fibers

Volume 79, number 5

OPTICS C O M M U N I C A T I O N S

1 November 1990

Dark solitons produced by phase steps in nonlinear optical fibers Yu.S. K i v s h a r a n d S.A. G r e d e s k u l Institute for Low Temperature Physics and Engineering, Ukr SSR Academy of Sciences, 47 Lenin Avenue Kharkov 310164, USSR Received 5 July 1989; revised manuscript received 21 June 1990

We demonstrate analytically that dark solitons may be produced from phase steps in nonlinear optical fibers at the positive group velocity dispersion. We study the cases of a step and two steps in the phase o f a cw background in detail. The parameters of generated solitons are calculated. The influence of the background broadening and dissipative losses on the dark-pulse evolution is discussed too.

1. Introduction

Although solitons arise in many areas of physics, the single-mode optical fibers have been found as most convenient one-dimensional objects to investigate properties of solitons. As was shown by Hasegawa and Tappert [ 1 ], in the region of the negative group-velocity dispersion ( G V D ) of a fiber it is possible propagation of so-called bright solitons. In this case the intensity at the pulse edges (at t ~ + oo) tends to zero. Since then, soliton propagation of bright optical pulses has been verified in a number of elegant experiments performed in the negative GVD region of the fiber spectrum (e.g., for 20> 1.3 gm in the case of silica, 20 being a wavelength) (see, e.g., a pioneer work by Mollenauer et al. [ 2 ] ). Most recently, transmission of 55-ps optical pulses through 6000 km of fiber was achieved by use of a combination of nonlinear soliton propagation and Raman amplification [31. For the positive GVD there are no bright solitons, instead the pulses undergo enhanced broadening and chirping. But in the positive GVD region there are so-called dark solitons, consisting of rapid dips in the intensity of a cw background, the latter are stable in this case. The existence of the solitons was theoretically predicted by Hasegawa and Tappert [4], but only recently dark solitons were observed in a number of experimental works [5-7]. In the experiments, dark solitons were observed on a broad bright pulse with a rapid intensity dip stipulated by a driv-

ing pulse [6], or utilizating a specially shaped antisymmetric input pulse of finite extent [5,7]. But as was demonstrated in ref. [ 8 ] and [ 9 ], the finite duration of the background is not the principal limitation for the dark-soliton propagation, and the pulses have similar properties as solitons for a cw background. Dark-soliton generation demonstrates a number of features, for example, unlike bright solitons [ 10 ], dark solitons may be generated without a power threshold [ 11 ]. This is the principal result for using the solitons in optical communication systems. The paper aims to demonstrate a new feature of dark-soliton creation, namely, dark solitons may be generated from phase steps of a background, i.e. only due to a phase modulation. The generation has no threshold for the first phase step, but has thresholds to create each additional dark soliton. The paper is organized as follows. In section 2 we present preliminary formulas describing the model under consideration, and some results related to the inverse scattering technique in the case of dark solitons. Section 3 is devoted to a single phase step, and section 4 describes the case of the soliton creation by two phase steps. In section 5 we briefly discuss the influence of the background broadening on the darksoliton propagation, and section 6 summarizes our results.

0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

285

Volume 79, number 5

OPTICS COMMUNICATIONS

2. Dark solitons and the inverse scattering transform

(O/at) ~1 = i)~, (o/at)q=

The propagation of short optical pulses in singlemode optical fibers is described by the well-known nonlinear Schr6dinger (NLS) equation [ 1,4]. In an appropriate system of normalized coordinates, this equation is Ou O2u i ~ zz - c y ~02 -+21u12u=0,

{1)

where u is the (complex) amplitude envelope of the pulse, z is the distance along the fiber, and the time variable t is a retarded time measured in a frame of reference moving along the fiber at the group velocity. The solutions of this equation divide into two different classes depending on the sign of a. In the case a = - I (negative G V D ) the appropriate b o u n d a ~ conditions in eq. ( 1 ) are I u l ~ 0 at t-~ _+or. For these conditions Zakharov and Shabat [12] showed that eq. ( 1 ) is an exactly integrable equation and possesses bright soliton solutions. At a = +1 (positive G V D ) the solution I u I ~ u0 = const is stable, and, as a result, eq. ( 1 ) has soliton solutions in the form of localized nonlinear "dark" excitations of the cw background. The NLS equation with the boundary conditions I u l ~ u0 = const, is also exactly integrable [ 13 ] and its one-soliton solution as an excitation of the cw background (the so-called dark soliton) has the general form (£-iu)2+exp(Z) U(z, t ) = u 0 , 1+ e x p ( Z )

I November 1990

exp(2iuZz+iOo

{2)

where

- i u ( 0 , t) ~ ,

(4a)

-iAT2 +iu*(0, t) ~1,

(4b)

where the asterisk means complex conjugation. As was shown by Zakharov and Shabat [ 13 ], each real discrete eigenvalue 121
3. A phase step The experiment by Weiner et al. [ 7 ] utilised a specially shaped anti-symmetric input pulse which closely related to the form of the fundamental (quiescent) dark soliton. The pulse has a phase difference at its edges which equals to ~. In this section we consider a more simple input pulse in the form of an arbitrary phase step and demonstrate that a dark soliton is always generated in this case. Let us consider a phase step on a cw background and take the input pulse in the form

(3)

u(0, t ) = u o e x p ( i a ) ,

t<0,

which corresponds to the boundary conditions lul-~uo at t-~_+~; the solution (2), (3) has only parameter u which characterizes the soliton intensity, 0o and to being arbitrary constant parameters. Let us consider eq. (1) with the boundary condition l u l ~ U o at t ~ + oo. According to the inverse scattering transform for eq. ( 1 ) at a = + 1 (see ref. [ 1 3 ] ) , to find which type of input pulse U ( z = 0 , t) generates solitons one has to investigate the eigenvalue Zakharov-Shabat (ZSh) problem [13]

= Uo exp(i//),

1>0.

Z=2uuo(t-to-2£uoz),

286

u = x / l - 2 "2,

(5)

The straightforward analysis of the ZSh spectral problem (4) with the "initial" condition (5) leads to the eigenfunctions of the discrete spectrum, { uoexp(ia) )

~-(t,x)=A-\~+i~ j X e x p [ t x / u o - 2 - ],

t<0,

(6a)

Volume 79, number 5

OPTICS COMMUNICATIONS

{ uoexp(ifl) "~ ~v+ (t, 2 ) = ~ + ~,2- i ~ , ] ×exp[-tx/~-~2

],

t>0,

(6b)

A_ and A+ being constant parameters. Using the matching condition, ~v_ (0, 2 ) = ~ + ( 0 , 2), we may find an equation.for the discrete spectrum and, as a result, its single real solution in the form 2 = - - U o cos ~0, q~= ½(fl--Ot) .

(7)

As was mentioned in section 2, the eigenvalue (7) corresponds to a dark soliton (2) with the " d a r k " intensity Uolsin ~01 and the velocity (the relative velocity in the t-space) - 2 U o cos ~0. In the case fl= a + n(2n + 1 ) we have a so-called "black" soliton

u(z, t)=-u(t)=uotanh(uo t) exp(2iu2z) ,

(8)

i.e. the fundamental pulse with the zero intensity at its center. In another case we have a "grey" soliton which has a lower-contrast intensity. Dark solitons produced by phase steps with the values ~0and ~r-q~ have the same intensities but opposite velocities. So, we have demonstrated analytically that a single phase step in the phase o f a cw background always produces a dark soliton; parameters o f the soliton are related directly to the phase difference 2~0. The results may be useful for the generation of dark solitons by the rather simple method using a phase modulation of the background.

1 November 1990

o f two phase steps on a cw background. Let us introduce the new variable ~0 [ cf. eq. (7) ]: 2 = - Uo cos ~0, 0 < ~0< n, then ~ o 2 - 22 = Uo sin ~0. For two steps when fl(t)=fl~O(t-t~)+fl20(t-t2), we have to find a solution as the solution (6a) for t t2, and a linear combination of simple solutions of the problem (4) at the phase fl= fl~ in the region t~ < t < t2. The matching conditions at the points t = tt and t = t2 yield the equation for eigenvalues, i.e. the equation for ~, sin (fl, / 2 - ~0) sin (f12/ 2 - ~0) = s i n ( i l l / 2 ) sin(fl2/2) e x p ( - 2 u 0 z sin q~)

where z - t 2 - t l ( z > 0 ) . It is interesting to note that the result for a single step obtained above follows directly from eq. (10): i f f l 2 = 0 then ~o=fll/2. In the case uorlsin ~01 >> 1, eq. (10) has two independent solutions which describe two dark solitons. Now we consider three different cases which demonstrate all features o f the dark-soliton generation by the step-modulated phase in detail.

Case fll=fl, f12=fl+27rn. Eq. (10) may be transformed into the following simple equation, sin2(fl/2-q~)=sin2(fl/2) exp( - 2 U o r sin ~0), ( 11 ) which has one or two real solutions. For r >> 1 we have two solutions,

q~.2=fl/2+sin(fl/2) exp[-uorsin(fl/2)] , 4. Two phase steps

and for ~--,0 we have the only solution,

To obtain a number o f dark solitons by the same way, we need to prepare a cw background with a variable phase, u(0, t ) = U o e x p [ i f l ( t ) ] , where, for simplicity, the f u n c t i o n / / ( t ) describes a number of steps (for simplicity, o r = 0 ) ,

~o,=fl-2Uorsin2(fl/2),

N

fl(t) = ~ fljO(t-tj),

(9)

(10)

(12)

for 0 < f l < ~ z ,

or

~ol=fl-n+2Uorsin2(fl/2),

for z r < f l < 2 g .

(13)

Direct analysis of eq. ( 1 1 ) yields the threshold value of r at which the second solution will arise,

j=l

fi
V,hr-----Uff~ Icotan(fl/2) I .

(14)

Therefore, for T< Tthr tWO equal phase steps generate only one dark soliton corresponding to the value 2 1 = - Uo sin ~0~, where ~01 is the unique solution of eq. ( 11 ); in the limit r>> rthr the solution has the form (13). But for r>zthr two dark solitons arise, and 287

Volume 79, number 5

OPTICS C O M M U N I C A T I O N S

above the threshold the second soliton corresponds to the following solution o f eq. ( 11 ), ~2 =2Uo('r--rmr) s i n 2 ( f l / 2 ) ,

1 November 1990

that its velocity is negative. In the opposite case the soliton velocity is positive.

for0
5. Finite-extent background

or

@2=Tg--2b/0(2"--~'thr) sin2(fl/2),

for~
The solution ( 1 5 ) is valid only near sin ~02= 0, i.e. for T-- ~Tthr <~ ~'thr" In the limit r>> rmr the solutions have equal parameters because in this limit we obtain r,01 ~ 0 2 = f l / 2 ,

(16)

0<~,.2
At last, we note that amplitudes of the generated dark solitons and their velocities are related to the values ~0 as Uol sin ~01 and - 2 U o cos ~o, respectively. Case fll = ,8, fll + fie = 2gn. Eq. ( 10 ) takes the form cos (2~0) - c o s f l = ( 1 - cos fl) exp( - 2Uor sin ~0) and always has two s y m m e t r i c solutions ~01 and ~02- n-~o~ (0<~o~ < n ) . In particular, at r ~ o e we obtain ~o1= f l / 2 and ~02= n - f l / 2 , i.e. two i n d e p e n d e n t solitons. In the opposite case when t-,O, we obtain the solution ~0~= 2 u o r sin2(fl/2) which m a y be also found by means o f the p e r t u r b a t i o n theory [9,1 1 ].

In experiments by Emplit et al. [5 ], Kr6kel et al. [6], and Weiner et al. [7], d a r k solitons were created on a background o f finite extent, the latter had the form o f a large bright pulse. But as was demonstrated in ref. [8] by direct numerical calculations, the dark pulses on a background o f finite extent have properties which are mostly similar to that o f exact dark solitons (i.e. dark solitons on a cw b a c k g r o u n d ) . In particular, as was found in ref. [ 8 ], the value Isz 2 is a p p r o x i m a t e l y equal to a constant along the fiber. Here Is is the dark-soliton intensity and rs is the soliton duration. This result has a simple theoretical explanation based on the asymptotic b e h a v i o u r of nonlinear non-soliton wavepackets [ 14]. As is well known, the bright pulses in the region o f the positive G V D undergo a b r o a d e n i n g due to the dispersion along the fiber. The evolution o f a bright pulse (in fact, the background for dark pulses) may be described in the framework o f the inverse scattering transform by an effective non-soliton wavepacket, and its asymptotic intensity m a y be presented in the form ( [ 14], see also c o m m e n t s in ref. [91): lu(z,l)12~(4gz)

General case. In a general case we have no exact analytical solutions o f eq. (10) but we can investigate the a p p e a r a n c e o f an a d d i t i o n a l (second) solution at the threshold, when the discrete eigenvalues are generated from the edges of the spectrum (2=-+Uo). The p e r t u r b a t i o n theory series near the critical points 2 = _+Uo (or, in terms o f ~o, near the points stipulated by the equation sin ~0= 0 ) allows us to find the threshold condition in a general form. The results are as follows. If fit +f124=27rn (in the opposite case we always have two solutions, the case is described a b o v e ) , an a d d i t i o n a l dark soliton arises when r > "rcr, .

z,t>>l,

where a ( 2 ) is the scattering a m p l i t u d e used in the inverse scattering transform. As we can see from eq. (18), l n [ z l 2 = O ( 1 ) , so that for a very large (but finite) background we may simply estimate Uo~ z - ~/2. Therefore, according to the solution (2 and ( 3 ) , the d u r a t i o n of the dark soliton is p r o p o r t i o n a l to Ts(UoP )

19)

1 ~zl/2,

and its m a x i m u m intensity, defined max (u 2 _ I u 12 ) is p r o p o r t i o n a l to '

as

Is= (20)

(17)

Moreover, if cotan (fix/2 ) + cotan (fl2/2) > 0, the new dark soliton is generated near the edge 2 = -Uo, so 288

2,

(18)

Is~bl2P2~Z

rcr= (2Uo) -~ Icotan(fl~/2) + c o t a n ( f l 2 / 2 ) l

'lnla(-t/4z)l

As a result, the value Isr 2 is not d e p e n d e n t on z. The same result was obtained numerically in ref. [8] in the case of a dark pulse produced on a background

Volume 79, number 5

OPTICS COMMUNICATIONS

o f finite extent. The results ( 1 9 ) a n d (20) estimate analytically the changing o f the soliton p a r a m e t e r s along a fiber due to dispersion b r o a d e n i n g o f the background. In real optical fibers solitons propagate in the presence o f dissipative losses. The effect m a y be described by the a d d i t i o n a l term - i y u in the rhs o f eq. (1). The dissipative loss decreases the intensity o f bright solitons along a fiber. Indeed, the bright soliton is a solution of eq. ( 1 ) at a = - 1,

u(x, t) =a exp(iaZx) /cosh(at) ,

(21)

[the soliton ( 2 1 ) is at rest in the reference frame moving with the group velocity]. Directly using the p e r t u r b a t i o n theory for the soliton (21 ) to take account of the dissipative term - i y u in the rhs o f eq. ( 1 ) (see, e.g., refs. [ 15,16 ] ), we m a y obtain the pert u r b a t i o n - i n d u c e d equation for the total field intensity

I= i --

[ul2 dt=2a'

dl

dz - - 2 y l ,

1 November 1990

between them. In the case o f two steps we have found the threshold value o f the d u r a t i o n exactly. As a result, the c o n d i t i o n t j + j - t j > > (2u~J)lcotan(flj+~/2)+ c o t a n ( f l J 2 ) I for the steps with the numbers j + 1 and j and the phase differences flj+l and/~j always quarantees the a p p e a r a n c e o f an a d d i t i o n a l ( j + 1 )th dark soliton. This result m a y be useful for the p r o d u c t i o n o f dark solitons due to a phase modulation. In the presence o f dispersive b r o a d e n i n g o f the finite-extent background, dark pulses adiabatically m a i n t a i n their m a i n soliton characteristics and the value Isz 2 is not d e p e n d e n t on z as the background intensity decreases along the fiber. This result m a y be o b t a i n e d by a very simple a p p r o a c h using the dependences o f the soliton p a r a m e t e r s on the background intensity and the asymptotic formula for nonsoliton wavepackets. We have also d e m o n s t r a t e d analytically that d a r k solitons spread more slowly than bright ones.

oc~

which yields the spatial evolution o f the soliton amplitude, a = a (0) exp ( - 2yz ). Therefore, for the width o f the bright soliton Zbs the inverse law is valid [see eq. (21)1,

% s = a - ~ ~ a - ~ ( O ) exp(2~z) . F o r dark solitons we m a y use the same a p p r o a c h as above in the case o f dispersion b r o a d e n i n g o f the background pulse. In the f r a m e w o r k o f the a p p r o a c h we m a y take into account the b a c k g r o u n d evolution in the linear a p p r o x i m a t i o n , Uo= Uo( 0 ) exp ( - ~,z). As a result, the evolution o f the dark-soliton width in the adiabatic a p p r o a c h is described as follows: r d s ~ ( U o V ) - ~ ~ c o n s t × e x p ( y z ) . It is obvious that dark solitons spread m o r e slowly than bright ones. The same conclusion was o b t a i n e d numerically in ref. [17].

6. Conclusions In conclusion, we have d e m o n s t r a t e d that steps in a cw-background phase p r o d u c e d a r k solitons. The quantity o f the solitons is exactly equal to the quantity o f the steps when the steps have a large d u r a t i o n

Acknowledgements The authors would like to thank the reviewer for useful comments.

References [ 1] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23 ( 1973 ) 142. [2] LF. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Len. 45 (1980) 1095. [ 3 ] L.F. Mollenauer and K. Smith, Optics Len. 13 (1988) 675. [4] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23 (1973) 171. [5] P. Emplit, J.P. Hamaide, F. Reynand, G. Froehly and A. Barthelemy, Optics Comm. 62 ( 1987 ) 374. [6] D. Kr/Akel,N.J. Halas, G. Guiliani and D. Grischkowsky, Phys. Rev. Lett. 60 (1988) 29. [7] A.M. Weiner, J.P. Heritage, R.J. Hawkins, R.N. Thurson, F.M. Kirschner, D.E. Leaird and W.J. Tomlinson, Phys. Rev. Lett 61 (1988) 2445. [8 ] W.J. Tomlinson, R.J. Hawkins, A.M. Weiner, J.P. Heritage and R.N. Thurston, J. Opt. Soc. Am. B 6 ( 1989 ) 329. [9] S.A. Gredeskul, Yu.S. Kivshar an M.V. Yanovskaya, Phys. Rev. A41 (1990) 3994. [ 10] Yu.S. Kivshar, J. Phys. A 22 (1989) 337. 289

Volume 79, number 5

OPTICS COMMUNICATIONS

[ 1 I ] S.A. Gredskul and Yu.S. Kivshar, Phys. Rev. Letl. 62 ( 1989 ) 977; Optics Lett. 14 (1989) 1281. [ 12] V.E. Zakharov and A.B. Sbabat, Sov. Phys. JETP 34 (1972) 62. [ 13 ] V.E. Zakharov and A.B. Shabat, Soy. Phys. JETP 37 ( 1973 ) 823.

290

I November 1990

[14]V.E. Zakharov, S.V. Manakov, S.P. Novikov and L.P. Pitaevsky, Theory of Solitons. The Inverse Scattering Transform (Nauka, Moscow, 1980). [ 15]A. Hasegawa, Optical solitons in fibers (Springer-Verlag, Berlin, 1989). [16]Yu.S. Kivshar and B.A. Malomed, Rev. Mod. Phys. 61 (1989) 763. [ 17] W. Zhao and E. Bourkoff, Optics Lett. 14 (1989) 703.