IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 26. NO. 12, DECEMBER 1990 2109

The Propagation of Bright and Dark Solitons in Lossy Optical Fibers

JUDITH A. GIANNINI, MEMBER, IEEE, AND RICHARD I. JOSEPH

Abstract-We develop an analytic perturbation solution to the non- linear Schrodinger equation with loss r for both normal and anoma- lous dispersion. Explicit results are obtained through second order in the perturbation r. Our results show that the dark pulse spreads less rapidly than the bright one and total spreading as well as the difference in spreading rate for the two types of pulses decreases with loss. Com- parisons are made with a zero order perturbation theory and with nu- merical simulations which are found to bracket our second order re- sults.

I. INTRODUCTION T is well known that narrow pulses propagating in op- I tical fibers experience temporal broadening resulting

from the dispersive properties of the fiber. In the presence of an intensity-dependent nonlinearity in the refractive in- dex ( n = no + $ n2 I E 1 2 ) , spectral broadening of the pulse can be used to counteract the effects of dispersion pro- ducing stable pulses that propagate without change in the absence of loss. These pulses are represented as stationary solutions to the nonlinear Schrodinger equation (as de- fined in [ l]) taking the form of the familiar bright soliton V I

1qI2 = A2 sech2 - ( t - L:, for negative dispersion and the dark soliton [2]

for positive dispersion, where q is the electric field am- plitude, z is the axial distance down the fiber, t is time, A2 is the maximum field intensity, 70 is the pulse half- width, and vg is the transmission speed of the pulse.

Bright soliton propagation was first verified by Mollen- auer et al. [3] in single-mode low-loss (0.2 dB/km) op- tical fiber over a short (700 m, effectively lossless ) dis- tance. More recently, Mollenauer and Smith [4] showed that with periodic Raman amplification bright soliton transmission could be extended to over 4000 km for sim- ilar fiber loss. The generation of dark pulses was first demonstrated by Emplit et al. [5] using amplitude and

Manuscript received March 12, 1990; revised July 28, 1990. J. A. Giannini is with the Applied Physics Laboratory, Johns Hopkins

R. I. Joseph is with the Department of Electrical and Computer Engi-

IEEE Log Number 9040281.

University, Laurel, MD 20723.

neering, Johns Hopkins University, Baltimore, MD 21218.

phase filtering. The formation of dark solitons in the nor- mal dispersion regime was verified by Krokel et al. [6] for a narrow dark pulse superimposed on a wide Gauss- ian-shaped background pulse. Their results showed the dark portion of the waveform propagated as a soliton while the Gaussian background spread. Finally, Weiner et al. [7] used a spatial mask with temporally nondispersive lenses and grating apparatus to construct a single, narrow, dark pulse with the appropriate phase change at the center demonstrating soliton propagation over a short (1.4 m, effectively lossless) fiber.

Attempts in the past to describe the effect of loss on soliton propagation for the most part have been the result of numerical integration of the nonlinear Schrodinger equation with (1) or (2) at z = 0 as the initial condition using techniques such as the split-step Fourier algorithm [8] or the propagating beam method [9]. For example, Doran and Blow [lo] numerically examined the effect of loss on the amount of spreading in the waveform for an initial hyperbolic secant shaped pulse. They showed that for all soliton modes, the spreading for the nonlinear pulse is less than for a linear pulse of the same shape. More recently, Zhao and Bourkoff [ 1 11 numerically examined the propagation properties of dark pulses with nondecay- ing background, studying their stability in the presence of loss and noise. They compared bright and dark pulse properties showing that the dark pulse spreads more slowly with loss and is less sensitive to noise than a bright pulse.

Unlike the previous works, Hasegawa and Kodama [12] used a perturbation scheme to evaluate the effect of loss on the bright pulse by expanding the scaled electric field amplitude as a power of the scaled loss ( r ) in the fiber. They show an expression for the pulse amplitude q to first order in I?. Note that the expression for q"' in (3.13) of [ 121 has a factor of 1 /2 that should be unity but this factor is believed to be of minor significance to the problem. Their expression for pulse spreading is based only on the zero order amplitude term. This is equivalent to assuming the initial amplitude phase is zero since the amplitude has only a real part. The zero phase assumption in deriving the spreading is necessary in their case because their so- liton, valid to first order in r, prevents them from solving for the energy condition which depends on r2 thus re- quiring the contribution from the second order perturba- tion term. As a result of this assumption, the power only

0018-9197/90/1200-2109$01.00 0 1990 IEEE

2110 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 26, NO. 12, DECEMBER 1990

of the zero order part of the solution decays as exp ( - 2 r t ) as it propagates. ( 5 = l O P 9 z / X where X is the wavelength of the light. )

In this paper, we derive expressions showing the effect of loss to second order in r for both bright and dark soliton solutions. The perturbation scheme used enables us to solve exactly for the initial phase of the pulse thus avoiding the difficulty encountered in [12] and allowing the total power to decay as required.

11. PERTURBATION SOLUTION The evolution of the electric field amplitude envelope

in the presence of loss is given by the nonlinear Schro- dinger equation as

2 iaq/d( + iDa2q/a27 + qIq( = -irq (3)

where D = f 1. As defined in [4]

5 = 1 0 - ~ 4 ~ (4)

= [ 1 0 - ~ . ~ / ( - A P ) ’ / ~ ] ( t - +,) ( 5 )

= 104.5(Tn2)1/2+ ( 6 )

r = 109xy (7) where 4 is the electric field envelope in a monomode fi- ber, n2 is the Kerr coefficient, k” is the dispersion in the fiber, and y is the fiber loss in dB/km. When y = 0, for anomalous dispersion (k” < 0, D = + 1 ) and for normal dispersion (k” > 0, D = - 1 ), (3) yields the usual bright and dark soliton solutions, respectively.

Making the transformations 7 = A exp ( -art) and q = 7Q in (3), then changing coordinates to x = 77 and y = r/q2 yields

iy[Q( 1 - U ) - maQ/dx + 2ayaQ/ay

+ iDa2Q/ax2 + QlQl’] = 0 (8)

where i is the f i and Q is written in terms of a complex amplitude and phase as

Q = exp [ i ( P + a/y)]F(x, Y) . (9)

The parameter a is a constant to be determined and /3 = - d 2 / I ’ . We construct a formal series solution for F of the form

m

~ ( x , Y) = n = O C Fn(x) (iy)”. (10)

Note that for A = 1, the phase of the waveform when I’ = 0 reduces to the phase for the usual lossless soliton solutions (i.e. - 4 for the bright soliton and - 14 for the dark soliton). For r # 0, the waveform is complex with a time dependent phase arising from the odd terms in the series (even at 5 = 0).

Substitution of (9)-(10) into (8) yields a set of equa- tions that are solved for the basis functions Fn, the first

few of which are

6DF;‘ + F l ( p + F i ) = ( a - 1)Fo + axF;, (12) /

4DF; + F2(p + 3F;) = FoF: - (1 + a)F , + UXF;

(13)

wherep = 2aa and the prime indicates differentiation with respect to the argument x.

A. Zero Order Solution The zero order equation yields the two usual stationary

bright and dark soliton solutions and defines the value re- quiredforp. F o r D = + l , p = -1/2and

Fo = sech (x) (14a)

= - 1 / 4 ~ . (14b)

Fo = tanh (x) (15a)

a = -1/2a. (15b)

ForD = - 1 , p = -1 and

Both (14) and (15) are stable in shape and constant in am- plitude as they propagate. The pulse spreading and am- plitude reduction of the Fo waveforms due to fiber loss are described by the F, and F2 (and higher order) terms in the series solution.

B. First Order Term The first order term in the series solution is obtained by

substituting FI = Fo G into (12) and eliminating F{ using (1 1) to give

i D ( F i G ) ’ = F o { ( a - 1)Fo + MF()} . (16)

Integrating twice gives

1 2

DF, = Fo (15 + - ax2 + K

where L and K are constants of integration that must be determined by conditions which depend on the specific choice of Fo (i.e., for D = + 1 or D = - 1 ).

C. Second Order Term The second order term of the series solution is obtained

by substituting F2 = FAH into (13) and eliminating F{ and F$ using (1 1) to give

&D[Fh2H’]‘ = FoFAF: - (1 + a)F;F; + mFAF,.

(18)

GIANNINI AND JOSEPH: PROPAGATION OF SOLITONS IN OPTICAL FIBERS 2111

Integrating twice gives

where N and M are constants of integration that, as be- fore, are determined by the choice of Fo.

111. BRIGHT SOLITON SOLUTION WITH Loss (D = + 1 ) The first perturbation term F , for the bright soliton case

is determined by evaluating the integrals in (17) upon sub- stitution of the zero order solution [( 1 l)] to give

F I = sech (x) ( L + lux' + $ K ( x + 1 sinh (2x))

(20) - 4(2 - a)[cosh (2x) - 1 1 ) .

The coefficients U and K are determined by requiring that F, -+ 0 in the limit where x + 03. Considering the be- havior of each of the terms in (20) in that limit yields: a = 2 and K = 0. Thus, (17) reduces to

F , = sech (x) { L + x 2 } (21 1 where L is to be determined by conditions on the total energy in the pulse once F2 is determined. (Note, from (14b), the phase factor in (9) is a! = - 1 /8. )

The second perturbation term F2 for the bright soliton case is determined by evaluating the integrals in (19) upon substitution of Fo [(14)] and F , [(21)]. Requiring that F2 --+ 0 in the limit where x --+ 00 yields: M = -2 In 2 and N = 0. Thus, (19) reduces to

4F2 = sech (x) tanh (x) { 3Z(x) + x(3 In 2 + 2L + 3 )

+ i x 3 ) - sech (x) (2L + x2 - +(x2 + L)'

+ 2 In [2 cosh (x) ] ] + tanh (x) ( -cosh (x)

+ sinh (x) In [ 2 cosh (x)] ] (22) where

Z(x) = s: d c l n [cosh ( c ) ] . (23)

One can now solve for L by requiring that the total en- ergy in the pulse decay exactly as exp ( -2rt ). The total

energy is defined as m

E = 1 dT(q ( ' (24) -CO

where from (9)-(10) and q = 9Q we have

1qI2 = qq* = q 2 { F i + y 2 [ F : - 2FOF21 + O(F4)} .

(25) Substituting (14), and (21)-(22) into (24)-(25) yields

dx sech2 (x) { 1 - y 2 2 } (26)

where

- (2L + x2 + 2 In [2 cosh (x) ] ]

+ tanh (x) [ - i x cosh ( 2 ~ )

(27) 1 1 2

+ - sinh (2x) In [ 2 (cosh (x)] .

Since the first term in (26) exactly integrates to exp ( -2I't), the second term in the integral must equal zero if E is to have the proper decay. Using this last condition, we can now solve for the last integration constant to get L = 1r2/24 and thus, E = 29 = 2 exp ( -2I'E) as re- quired.

Iv. DARK SOLITON SOLUTION WITH LOSS ( D = - 1 ) The first perturbation term F I in this case is determined

by evaluating the integrals in (17) upon substitution of (1 5) to give

-FI = Ltanh (x) + X ' ( U - 1 ) tanh (x)

+ K ( x tanh (x) - 1 ) + ~ ( 2 - U ) . (28)

The constants a and K are determined by requiring that F; -+ 0 in the limit where x -, 03. Considering the be- havior of each of the terms in (28) in that limit yields: a = 1 and K = 0. Thus, (28) reduces to

F1 = -L tanh (x) - x (29)

where L is to be determined by conditions on the differ- ence between the total pulse energy and the background once F2 is determined. (Note, from (15b), the phase fac- tor in (9) is a! = - 1 /2. )

The second perturbation term F2 for the dark soliton case is determined by evaluating the integrals in (19) upon substitution of Fo [(15)] and F I [(29)] and requiring as before that F2 + 0 in the limit as x -+ 03. With these conditions, it can be shown that M = -2 In 2 and N = 0

2112 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 26. NO. 12, DECEMBER 1990

which yields

2F2 = sech2 (x) { 31 + x(3 In 2 + L + 3) + 4x3}

+ cosh (x) { -2x cosh (x)

+ 2 sinh ( x ) In [2 cosh (x)]]

+ tanh (x)(2L - x + (x + Lf

+ 3 In [2 cosh ( x ) ] ] (30) where Z(x) is defined as in (23).

The value of L is now determined by requiring that the total energy in the pulse minus the background energy de- cays as exp ( -FE). This energy difference is defined as

P -

E = J --OD dr{ -M2 + 141')

where 1 q l 2 is defined as in (25). The background levels, obtained by examining the asymptotic behavior of each of the terms Fo, F 1 , and F2 as x -+ 00, are given as

- Fo = 1 (32)

F 1 - - - L - x (33) - F2 = +x2 + x(4 + L) + +L2 + L + (34)

such that

+ I 2 + ) I 2 = (-F2, + F2,) + 6: (35) where

d: = ( - F : + F : ) - (-2FoF1 + 2FoF1). (36)

Substituting ( 1 9 , (29)-(30), and (32)-(35) into (25), and (31) yields

E = 27 dr{ (-Fi + F i ) - y26:}. (37) lorn As before, the integral of 6: must equal zero for E to de- cay as required. Using this condition, solving for L gives L = (7r2/12) + and E = 2 exp (-F[).

V. RESULTS To illustrate the effect of loss, the intensities of the

bright and dark pulses were computed varying both the propagation distance and the fiber loss. The bright pulse intensity [ I q l 2 as given in (25)] and the dark pulse inten- sity with the background removed [ - I qI2 + I i f ( ' as given in (35)] are plotted as a function of the reduced time T [(5)] in picoseconds for several values of the normalized propagation distance ,$ . Conversion to kilometers can be made using (4) with X in kilometers. In what follows, we will use as typical values for the dimensionless loss I' = 0.02993 and 0.01497. Conversion to loss in dB/km can be made as

20 1 y(dB/km) = I' * - - ~

In 10 h ( p m ) '

1 I\

AT/ATO 11 BRIGHTPULSE - 0 10000 r - n n+ 407

"- 4 11249 ; 24 19958

48 3.1626

. .. .

- "." ,-e,

5 0.4

z I \ 0.0 2 4 6 8 10 12 14 16 18 20

TAU

Fig. 1 . Evolution of the bright pulse intensity profile as a function of nor- malized propagation distance for a loss of r = 0.01497. The solid line indicates the pulse shape at the start of propagation ( t = 0 ) . A T / A TO indicates the pulse full width at half maximum ratio in picoseconds at each [ value. For this loss the pulse has doubled its width in = 23.

Thus, at X = 1.3 pm and I' = 0.02993, y = 0.2 dB/km which corresponds to the values used by Hasegawa and Kodama [12]. Note that the values y = 0.2 dB/km, I' = 0.03567, and X = 1.55 pm correspond to the parameters in the bright soliton experiment in [3]. The values y = 0.2 dB/km, I' = 0.01428 and X = 0.62 pm correspond to the parameters in the dark soliton experiment in [7].

Consider first the bright soliton case. For a loss I' = 0.01497 (Fig. l), the pulse is shown to double its width ( A T = full width at half maximum) after ,$ = 23. In that distance, the peak amplitude has been attenuated by over 75 % . From the form of the zero order solution, it is seen that the maximum intensity decays as q2 = exp ( - 4 Q ) while the pulse spreads as 1 / q = exp (2r.g ). This indi- cates that the product of loss and pulsewidth doubling dis- tance is constant (In 2/2). The constancy in this product is a natural consequence of the length scale over which the energy in the pulse decays [(26)]. Thus for a loss 1 / 10 that in Fig. 1 (I' = 0.001497), the doubling distance in- creases by a factor of 10 to [ = 232. For the higher values of I' = 0.02993 and 0.03569 (in [3]), the doubling dis- tances ,$ are 12 and 10, respectively. This last value is compared with the maximum propagation distance for the experiment ,$ = 0.45 ( z = 700 m at X = 1.55 pm) indi- cating that, as observed, significant pulse spreading is not expected.

Similar results are observed for the dark soliton. Note that for a loss of 0.01497 (Fig. 2), the dark pulse doubles its width slower (in [ = 46) than the corresponding bright pulse. (The doubling distances for r = 0.02993 and 0.01428 (in [7]) are [ = 32 and 49, respectively. This last value is compared with the maximum propagation distance in the experiment [7] [ = 2.26 - ( z = 1.4 m at X = 0.62 pm) again indicating no significant pulse spreading. From the form of the zero order solution, it is seen that the maximum intensity decays as q2 = exp ( - 2I'[ ) while the pulse spreads as 1 / q = exp ( I'q ). This indicates that the product of loss and doubling distance is

21 13 GIANNINI AND JOSEPH: PROPAGATION OF SOLITONS IN OPTICAL FIBERS

4 .

48 20759 I , ' I 1 , 1 ,

w - - - - C O 4 - ; 1 .

9 g o 2 . - - - - - -___

\ \ \

0 ' " " 0 0 2 4 6 8 10 12 14 16 18 20

TAU

Fig 2 Evolution of the dark pulse intensity for a loss of 0.01497 with the background intensity removed. As for the bnght pulse (Fig l ) , the solid line indicates the pulse shape at the start of propagation ( E = 0) A T / A TO indicates the dark pulse has doubled its width slower than the corresponding bnght pulse (in E = 47) .

- - - - - DARKPULSE

BRIGHT PULSE

constant (In 2) and as for the bright pulse, if the loss is decreased by a factor of 10, the doubling distance in- creases by a factor of 10 to [ 2: 480 in this case. Plots of the pulsewidth ratio [ A T ( [ ) / A T ( [ = 0 ) ] for the bright and dark solutions (Fig. 3) emphasize this behavior. At the highest loss (0.02993), there is a noticeable deviation ( = 12 % ) between the bright and dark pulsewidths at rel- atively small distances ( E = 4) . As expected, the slope of the pulsewidth ratio decreases with loss and the cor- responding curves for bright and dark pulses at a given loss agree more closely at larger distances. At the lowest loss plotted (0.002993 ) the deviation between the spread- ing ratio of the two solutions reaches 12% at E = 40.

A comparison of this ratio for the second order bright pulse theory (solid line) with the zero order prediction of Hasegawa and Kodama [12] (dashed line) at r = 0.01497 (Fig. 4) shows agreement at the smaller values of [. The divergence of the two curves is due primarily to the dif- ference in order of the solutions. (Note that Hasegawa and Kodama use only a zero order term which is purely real and corresponds to zero phase.) Also, at larger [ the sec- ond order theory begins to break down due to the series nature of the solution which requires that the expansion parameter ( y = r/V2) be much less than unity for con- vergence. The value of y producing adequate convergence was determined by comparing the theory with a numerical simulation and examining the shape of the pulsewidth ra- tio curve. For y = 0.1, the deviation between the theory and the numerical results was less than 5 % . At r = 0.01497 this y corresponds to [ = 32. An examination of the pulsewidth ratio for this loss shows that at [ 2: 32 there is a break in the curve as the second derivative changes sign at a small plateau before a rapid growth. This same behavior is observed at other values of r for the same y indicating convergence for the bright pulse with r = 0.01497 and 0.02993 to [ = 32 and 10, respectively. Similarly, the dark-pulse theory with r = 0.01497 and 0.02993 converges to [ = 79 and 28, respectively, with

I I I I I

0 10 20 30 40 SCALED DISTANCE t

Fig. 3 . Comparison of pulsewidth ratio ( A T ( [ ) / A T ( t = 0 ) for bright and dark pulses at three losses. The deviation between the bright and dark pulse curves as well as the slope of the curves decrease as loss decreases.

- GIANNlNl AND JOSEPH

- - - - HASEGAWA AND KODAhlA

. . . . . . . . W O AND BOURKOFF

BRIGHT PULSE

r = o 01497 .

0 10 20 30 40 SCALED DISTANCE t

Fig. 4 . Comparison of bright pulsewidth ratio at a loss of 0.01497 for the second order theory (solid line) with the zero order theory [12] (dashed line) and numerical computation [lo] (long dashed line) and [ I I ] (dotted line). The flattening in the solid line at [ greater than = 30 indicates the propagation distance where the perturbation is beginning to break down at this loss. (To correct for differences in the r value used, the for the numerical computations has been scaled by 0.05/0.01497. )

a value of y = 0.16. Fig. 4 shows that numerical com- putations by Doran and Blow [lo] (long dashed curve) and Zhao and Bourkoff [ l 11 (dotted curve) (both of whom use a real hyperbolic secant for their input) and the zero order theory of Hasegawa and Kodama bracket our sec- ond order results in the valid region of [ with the numer- ical computations showing the lowest estimate (agreeing to = 10% at [ = 30). The difference between the second order theory and the numerical results in [lo], [ l l ] is due to the initial phase on our input waveform. The T-dependent phase across the pulse causes the waveform initially to spread faster than the numerical results but our spreading rate (slope) becomes approximately equal to the numerical results in the valid region of [ after the pulse has propagated a short distance. It is assumed that the difference between the two numerical estimates is due to

21 14 IEEE J O U R N A L OF Q U A N T U M ELhC’IKONICS, VOL. 26, NO. 12, DECEMBER 1990

- GWJNlNl AND JOSEPH

- - - _ ZERO ORDER

. . . . . . . . zw\O AND BOURKOFF

DARK PULSE

r = 0.01 497

0 10 20 30 40

SCALED DISTANCE C

Fig. 5 . Comparison of dark pulsewidth ratio at a loss of 0.01497 for the second order theory (solid line) with the zero order theory (dashed line) and numerical computation [6] (dotted line). (To correct for differences in the r value used, the ( for the numerical computations has been scaled by 0.05/0.01497.)

differences in the implementation of the numerical tech- niques used.

Finally, a comparison of our second order dark pulse theory (solid line) with a numerical integration by Zhao and Bourkoff [ 113 (who uses a purely real hyperbolic tan- gent input) (dotted line) at r = 0.01497 (Fig. 5) shows that the numerical computations again give a lower esti- mate for the pulse spreading. The dashed curve in the fig- ure represents only the zero order term in our theory and corresponds to the zero order prediction of the pulse width of Hasegawa and Kodama for the bright pulse case. As shown, the contribution from the higher order terms for the dark pulse case for the k range shown is small. Based on the value of the expansion parameter y, a noticeable deviation between the zero order and second order theory will not be seen until 4 = 70. Like the results of Zhao and Bourkoff, our dark pulse spreads slower than the bright pulse with a deviation between the two that is com- parable to that seen in the bright case in Fig. 4.

VI. CONCLUSION We have developed an analytic solution for the nonlin-

ear Schrodinger equation with loss to second order in the perturbation parameter r for both the bright and the dark pulses. Our solution shows that of the two waveforms the dark pulse spreads and attenuates more slowly with a width doubling distance approximately twice that of the bright pulse indicating good agreement with numerical simulations [lo], [ 113 having purely real hyperbolic se- cant or hyperbolic tangent inputs. Our results indicate that a .r-dependent phase on the input pulse increases initial spreading resulting in greater absolute spreading during propagation than an equivalent pulse with a purely real amplitude.

REFERENCES [ I ] A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear

optical pulses in dispersive dielectric fibers. I. Anomalous disper- sion,” Appl. Phys. Le??., vol. 23, pp. 142-144, 1973.

[2] -, “Transmission of stationary nonlinear optical pulses in disper- sive dielectric fibers. 11. Normal dispersion,” Appl. Phys. Let t . , vol. 23, pp. 171-172, 1973.

[3] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fi- bers,” Phys. Rev. Le f t . , vol. 45, pp. 1095-1098, 1980.

[4] L. F. Mollenauer and K. Smith, “Demonstration of soliton transmis- sion over more than 4000 km in fiber with loss periodically compen- sated by Raman gain,” Opt. Let t . , vol. 13, pp. 675-677, 1988.

[ 5 ] P. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthe- lemy, “Picosecond steps and dark pulses through nonlinear single mode fibers,” Opt. Commun.,vol. 62, pp. 374-379, 1987.

[6] D. Krokel, N. J. Halas, G. Giuliani, and D. Gnschkowsky, “Dark- pulse propagation in optical fibers,” Phys. Rev. Lett., vol. 60, pp. 29-32, 1988.

171 A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Let t . , vol. 61, pp. 2445-2448, 1988.

[8] F. Tappert and C. Judice, “Recurrence of nonlinear ion acoustic waves,” Phys. Rev. Let t . , vol. 29, pp. 1308-131 1, 1972.

[9] D. Yevick and B. Hermansson, “Soliton analysis with the propagat- ing beam method,” Opt. Commun., vol. 47, pp. 101-106, 1983.

[ lo] N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron., vol. QE-19, pp. 1883-1888, 1983.

[ l l ] W. Zhao and E. Bourkoff, “Propagation properties of dark solitons,” Opt. Lett., vol. 14, pp. 703-705, 1989.

[12] A. Hasegawa and Y. Kodama, “Signal transmission by optical soli- tons in monomode fiber,” Proc. IEEE, vol. 69, pp. 1145-1 150, 1981.

Judith A. Giannini (M’80) received the B.S. and M.S. degrees, both in physics, from Drexel Uni- versity, Philadelphia, PA, in 1974 and 1976, re- spectively.

She joined the Applied Physics Laboratory, Johns Hopkins University, Laurel, MD, in 1977, where she is a member of the Senior Professional Staff. Her interests have included studying the physics of the ocean in the areas of electromag- netic propagation, remote sensing, and nonlinear phenomena. Currently, she is a Ph.D. candidate

with the Department of Electrical and Computer Engineering, Johns Hop- kins University. Her research area is soliton propagation in optical mate- rials.

Ms. Giannini is a member of the American Physical Society and the Optical Society of America.

Richard I. Joseph received the B.S. degree from City College of the City University of New York, New York, in 1957 and the Ph.D. degree from Harvard University, Cambridge, MA, in 1962, both in physics.

From 1961 to 1966, he was a senior scientist with the Research Division of the Raytheon Co. Since 1966, he has been with the Department of Electrical and Computer Engineering, Johns Hop- kins University, where he is currently the Jacob Suter Jammer Professor of Electrical and Com-

puter Engineering, as well as a member of the Principal Professional Staff of the Applied Physics Laboratory. During 1972, he was a Visiting Pro- fessor of Physics at Kings College, University of London, London, on a Guggenheim Fellowship.

Dr. Joseph is a Fellow of the American Physical Society.