Grandpierre et al. Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. B 55

Gray spatial solitons in biased photorefractivemedia

Alexandra G. Grandpierre and Demetrios N. Christodoulides

Department of Electrical Engineering and Computer Science, Lehigh University, Bethlehem, Pennsylvania 18015

Tamer H. Coskun

Department of Electrical and Electronics Engineering, Pamukkale University, Denizli, Turkey

Mordechai Segev

Department of Physics, Technion—Israel Institute of Technology, Haifa 32000, Israeland Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544

Yuri S. Kivshar

Optical Sciences Center, Research School of Physical Sciences and Engineering, Australian National University,Canberra ACT 0200, Australia

Received February 29, 2000; revised manuscript received September 11, 2000

We provide a detailed analysis of gray spatial optical solitons in biased photorefractive media. The propertiesassociated with these solitons, such as their transverse velocity, spatial width, and phase profile, are obtainedas functions of their normalized intensity and degree of ‘‘grayness.’’ By employing the stability criterion basedon the renormalized field momentum, we investigate the stability regions of gray spatial photorefractive soli-tons. The process of the soliton Y splitting arising from an initially even field depression is quasi-analyticallydescribed by use of a Hamiltonian formalism. © 2001 Optical Society of America

OCIS codes: 190.5330, 190.5530, 190.5940.

1. INTRODUCTIONDuring the past decade or so, dark optical solitons—bothtemporal and spatial—have been the focus of considerableattention.1 In the spatiotemporal domain, dark opticalsolitons were first observed in low-loss optical fibers withnormal dispersion.2,3 Subsequently, dark spatial opticalsolitons were experimentally demonstrated in self-defocusing nonlinear media.4–8 In this latter case a darksoliton or dark-soliton stripe (a dark notch or a void in in-tensity) forms when the effect of diffraction is exactly bal-anced by the self-defocusing nonlinearity of an optical me-dium. Thus far several physical systems have beenidentified as capable of supporting dark spatial solitons.These include Kerr media such as liquids, gases, andsemiconductors as well as photorefractive crystals.4–14

The case of photorefractive solitons is of particular in-terest, since a photorefractive crystal does not resemble aKerr material. In fact, photorefractive media are typi-cally anisotropic, and their nonlinearity is nonlocal. Yet,as has been suggested theoretically, both bright and darkspatial solitons are possible in either biased photo-refractives15–18 or in crystals with appreciable photovol-taic coefficients.19–21 The existence of steady-state pho-torefractive spatial solitons was successfully demon-strated in a series of experimental studies.22 Inparticular, dark spatial solitons in the form of darkstripes or vortices have been observed in reverse-biased

0740-3224/2001/010055-09$15.00 ©

strontium barium niobate photorefractive crystals andphotovoltaic lithium niobate samples.9–14 In all of thesecases, the observation of dark solitons was possible atrelatively low power levels. Moreover, it was shown thattheir soliton-induced waveguides are capable of guidingother, more powerful beams at less-photosensitivewavelengths.23

One of the most interesting features of a dark-solitonbeam is its ability to display a splitting behavior duringpropagation.12,24–27 Depending on its input amplitude–phase profile, a dark beam can split into a sequence ofdark or gray solitons. For example, under appropriateinitial conditions, a dark beam can generate an odd num-ber of dark depressions, provided its phase has a p phasejump at the input (odd phase profile),28–30 or it may ex-hibit a more complicated splitting behavior in a generalcase.31 When the phase jump is exactly p, the beam atthe center forms a dark (black) soliton, whereas the re-maining depressions represent pairs of gray solitons trav-eling in opposite directions. If, on the other hand, the in-put phase is constant across the dark beam (even phaseprofile), an even number of gray solitons emerge.27–29 Inthis latter case pairs of gray solitons are generated andtravel in opposite transverse directions so as to conservethe overall linear momentum of the system. This solitonY splitting process, which is unique to dark-solitonbeams, has been successfully used to write permanent

2001 Optical Society of America

56 J. Opt. Soc. Am. B/Vol. 18, No. 1 /January 2001 Grandpierre et al.

Y-junction waveguide structures (3 dB splitters) in thebulk of a photorefractive crystal.32,33 Such Y junctionscan then be employed to guide signal (or probe) beams atless-photosensitive wavelengths (;1.5 mm) and can bepermanently impressed or fixed into the crystalline lat-tice. Clearly, an investigation of the properties of thesesoliton-induced Y-junction photorefractive waveguides re-quires an in-depth understanding of the underlying soli-ton splitting dynamics. It is important to emphasizethat, even though gray solitons were previously consid-ered within the context of saturable Kerr nonlinearities,4

the case of gray photorefractive solitons is by nature morecomplicated and so far has remained unexplored. To bemore specific, dark beams in biased photorefractive crys-tals are governed by a nonlinear Schrodinger equation ofthe saturable type, which directly involves the boundaryconditions of the space-charge field through the so-called(1 1 r) factor.16–18 This, in turn, has important implica-tions for the properties or characteristics of these photo-refractive gray-soliton states. In this paper we provide adetailed analysis of gray spatial optical solitons in biasedphotorefractive media. The properties of these gray-soliton states, such as their transverse velocity, spatialwidth, and phase profile, are obtained as functions oftheir normalized intensity and their degree of grayness.In a certain range of the soliton parameters we find thatthe soliton total phase shift exceeds p, i.e., they become‘‘darker than black.’’34 Moreover, by employing the sta-bility criterion based on the renormalized momentum,35,36

we investigate the stability properties of gray photore-fractive solitons. The Y soliton splitting arising from aninitially even field depression is also semianalytically de-scribed by use of a Hamiltonian formalism and the asso-ciated conserved quantities. This, in turn, allows one topredict the Y-splitting angle and the grayness of the gen-erated soliton pair once the parameters of the even fielddepression have been specified.

2. PROBLEM FORMULATION:DEFINITIONSWe start our analysis by considering a planar (stripe) op-tical beam propagating in a photorefractive materialalong the z axis. Being planar, the beam is allowed todiffract only along the x direction. Without any loss ofgenerality, let us assume that the photorefractive crystalconsidered here is strontium barium niobate (SBN), withits optical c axis oriented along the x coordinate. The op-tical beam is linearly polarized along x, and the externalbias field is applied in the same direction. Under theseconditions it can be directly shown that the slowly vary-ing envelope f of the optical beam obeys the followingevolution equation16–18:

i]f

]z1

1

2k

]2f

]x22

k0

2~ne

3r33 Esc!f 5 0, (1)

where Esc is the space charge field induced in this photo-refractive material, which under steady-state conditionsis given by

Esc 5 E0S I` 1 Id

I 1 IdD . (2)

Here ne is the unperturbed extraordinary index of refrac-tion, k0 5 2p/l0 , k 5 k0ne , and r33 is the electro-opticcoefficient involved. Other quantities are I 5 I(x, z),the power density of the optical beam; Id , the dark irra-diance of the crystal; and I` , the intensity of the waveaway from the dark notch of the wave, i.e., I` 5 I(x→ 6`). E0 is the value of the space charge field at x→ 6`. If the spatial extent of the optical beam’s darkdepression is much less than the x-width W of the photo-refractive crystal, then E0 . 6V/W, where V is the exter-nal bias voltage used. Corrections regarding this ap-proximation (E0 . V/W) can be found in Appendix A.We also point out that the dark irradiance of the crystalId can be artificially elevated by proper illumination ofthe crystal as demonstrated in previous experimentalstudies.12

It proves more convenient to normalize Eq. (1) byadopting the following transformations and dimension-less coordinates: f 5 (2h0Id /ne)

1/2U, j 5 k0ne3r33uE0u

(z/2), and s 5 k0ne2(r33uE0u/2)1/2x. In this case one ob-

tains

i]U

]j1

1

2

]2U

]s21 ~1 1 r!

U

1 1 uUu25 0, (3)

where r 5 I` /Id is the so-called intensity ratio of thedark beam at x → 6` with respect to Id and j and s aredimensionless coordinates. The envelope U has beenscaled with respect to the dark irradiance with h0 repre-senting the free-space intrinsic impedance. It is impor-tant to note that in the case of dark beams the boundaryconditions of the space-charge field are directly involvedin the evolution equation through the (1 1 r) factor. Inessence, the strength of the nonlinearity increases lin-early with the dark intensity ratio r. In deriving Eq. (3)we have also implicitly assumed that E0 is negative sothat dark solitons can be supported through the inducedself-defocusing nonlinearity.

Equation (3) can be further renormalized1 by use of thefollowing gauge transformation U 5 Cexp(ij), in whichcase it takes the form:

i]C

]j1

1

2

]2C

]s21 F ~1 1 r!

1 1 uCu22 1GC 5 0. (4)

Note that in Eq. (4) the nonlinear term goes to zero ats → 6` for optical beams of the dark type, since uCu2

→ r for s → 6`. It is straightforward to show that Eq.(4) can also be obtained from the following Lagrangiandensity:

L 5i

2~CCj* 2 C* Cj! 1

1

2CsCs*

1 ~ uCu2 2 r! 1 ~1 1 r!lnF ~1 1 r!

1 1 uCu2G , (5)

where C* is the complex conjugate of C, Cj 5 ]C/]j, etc.Similarly, this system can be described within a Hamil-

Grandpierre et al. Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. B 57

tonian formalism. In this case, Eqs. (3) and (4) can be de-rived from the Hamiltonian density:

H 51

2uUsu2 1 ~1 1 r!lnF ~1 1 r!

1 1 uUu2G 1 ~ uUu2 2 r!, (6)

where in Eq. (6) we have used the fact that uCu2 5 uUu2

and uCsu 5 uUs u. One can also easily show that Eq. (3)exhibits the following conservation laws:

N 5 E2`

`

ds~ uUu2 2 r!, (7)

P 5i

2E

2`

`

ds~Us* U 2 U* Us!S 1 2r

uUu2D5

i

2E

2`

`

ds~Us* U 2 U* Us! 2 r Arg Uu2`` , (8)

H 5 E2`

`

dsH 1

2uUsu2 1 ~1 1 r!lnF ~1 1 r!

1 1 uUu2G1 ~ uUu2 2 r!J , (9)

where N is related to the complementary power conveyedby the optical beam, P is the so-called renormalized mo-mentum of the system, and H is the renormalized Hamil-tonian. All three quantities, N, P, and H are finite be-cause of this renormalization procedure, and, of course,they remain invariant during beam propagation. Inwhat follows, the gray soliton solutions of Eq. (3) will beobtained.

3. PROPERTIES OF GRAYPHOTOREFRACTIVE SOLITONSBefore considering the gray-soliton solution of Eq. (3), it isuseful to point out its Galilean invariance. In otherwords, one can show that after introduction of the trans-formation

U~s, j! 5 A~h,z!exp~iah!exp~ilz! (10)

in Eq. (3) and the moving coordinates h 5 s 2 vj, z5 j, the new envelope A(h,z) satisfies the same evolu-tion equation [that is Eq. (3) governing U],

i]A

]z1

1

2

]2A

]h21 ~1 1 r!

A

1 1 uAu25 0. (11)

Equation (11) is true provided that the gauge parametersa and l satisfy the conditions a 5 v and l 5 v2/2. Inthis new moving-coordinate system v plays the role of anormalized transverse soliton velocity. Because of thisone-to-one correspondence, any solution of Eq. (11) auto-matically satisfies Eq. (3) and vice versa. It can be di-rectly shown that the gray-soliton solutions of Eq. (11) aregiven by17

A 5 r1/2y~h!expF imz 2 iJE0

h dh8

y2~h8!1 iF0G , (12)

where J is a constant, y(h) is a normalized function(u y(h)u < 1), and F0 is an arbitrary phase. By imposingthe condition J 5 v, the solution for the U envelope nowreads as

U 5 r1/2y~h!expH iS m 1v2

2D z

1 ivFh 2 E0

h dh8

y2~h8!G 1 iF0J . (13)

The condition J 5 v was intentionally employed so as thephase of this gray spatial soliton is constant when h ors → 6`, which is consistent with the excitation condi-tions right at the origin. The quantity (m 1 v2/2) in Eq.(13) represents a nonlinear shift in the propagation con-stant. By substituting Eq. (13) into Eq. (3) we find thatthe normalized field profile y(h) satisfies the following or-dinary differential equation:

d2y

dh22 2my 2

v2

y31 ~1 1 r!

2y

1 1 ry25 0. (14)

We now look for gray-soliton solutions by imposing theboundary conditions, y(0) 5 Am and y(h) 5 1 at h5 6`. The m parameter (0 , m , 1) describes thesoliton grayness, i.e., the intensity at the middle of thewave defined as I(0) 5 mI` .

Equation (14) can be easily integrated once (by quadra-ture), leading to

~ y8!2

22 my2 1

v2

2y21

~1 1 r!

rln~1 1 ry2! 5 C, (15)

where y8 5 dy/dh and C is an integration constant. Byapplying the boundary conditions at h → 6` and at h5 0, one can determine the parameters m, v, and C.With the conditions y 5 1 and y8 5 y9 5 0 at h → 6`,and y2(0) 5 m, y8(0) 5 0, we get

C 5 2m 1v2

21

~1 1 r!

rln~1 1 r!, (16)

C 5 2mm 1v2

2m1

~1 1 r!

rln~1 1 mr!, (17)

v 5 @2~1 2 m!#1/2, (18)

from which the parameter m can be determined, i.e.,

m 51

1 2 m1

m

~1 2 m !2

~1 1 r!

rlnS 1 1 mr

1 1 rD . (19)

Thus, given the grayness m of this gray spatial soliton aswell as its maximum intensity ratio r, one can thenuniquely determine the parameter m from Eq. (19) and, inturn, its transverse velocity v from Eq. (18) and C fromeither Eqs. (16) or (17). From here one can obtain thenormalized field profile y(h) by numerically integratingEq. (15).

Figure 1(a) shows the normalized intensity of a grayphotorefractive spatial optical soliton as a function of s fordifferent values of the grayness parameter m when r

58 J. Opt. Soc. Am. B/Vol. 18, No. 1 /January 2001 Grandpierre et al.

5 40. The associated soliton phase profiles are given inFig. 1(b) for the same range of parameters. It is obviousfrom Fig. 1(a) that for a given intensity ratio the spatialsoliton becomes narrower as m decreases (i.e., as the soli-ton becomes darker).

The normalized transverse velocity v is depicted in Fig.2 as a function of r for various values of the grayness pa-rameter m. As one can see, the normalized velocity v isbounded between 0 and 1. One can easily show from Eq.(19) that as r → 0, m → 1, and thus v → 0. On the otherhand, the same equation indicates that as r → ` and m→ 1, the m parameter approaches 1/2, and therefore v→ 1. Moreover, the normalized velocity quickly in-creases in the neighborhood of r ' 0 and becomes almostconstant at higher values of r (r * 40). It is also inter-esting to note from Eqs. (18) and (19) that the solitontransverse velocity is approaching zero as m → 0. Thisshould have been anticipated, since this case correspondsto the fundamental dark (black) spatial soliton, the veloc-ity of which is zero.

The normalized FWHM of the gray-soliton beam Dh isshown in Fig. 3, where Dh is defined as the spatial FWHMbetween the background and the lowest value of the in-tensity, i.e., the points where y2(h) 5 (1 1 m)/2. Figure3 clearly indicates that the beam width generally de-creases with the normalized intensity ratio r. At verylow values of r (i.e., in the Kerr limit), Dh decreases veryrapidly, whereas it saturates for r * 40. This saturationis predominantly due to the fact that the evolution equa-

Fig. 1. (a) Normalized intensity profile of a gray photorefractivesoliton at r 5 40 and for various values of the grayness factor m;(b) associated phase profile of these gray solitons.

tion itself involves the space-charge-field boundary condi-tions through the (1 1 r) factor. In fact, had this effectnot been present, the FWHM curves would have behavedsimilarly to those of bright photorefractive solitons.16–18

The beam width also increases with the degree of gray-ness m.

As noted above, the antisymmetric (with respect to h)phase profile of a gray photorefractive soliton is shown inFig. 1(b). Of interest is the soliton total phase jump, i.e.,the phase difference between the two infinite tails of thelocalized structure, shown in Fig. 4 for various values ofr,m. As follows from this figure, this phase differencesaturates at high values of r. Note that the phase jumpexceeds p for relatively low degrees of grayness (when thesoliton is close to being black). In essence, in this regime(m & 0.2) these gray spatial solitons can be called‘‘darker than black,’’ and similar results were previouslyfound in the case of gray solitons in other saturable Kerrmedia.34

The angle u at which a spatial dark soliton propagateswith respect to the z axis is related to its transverse ve-locity v and in radians is given by

Fig. 2. Transverse velocity v of a gray photorefractive soliton asa function of r and m.

Fig. 3. Normalized FWHM of a gray photorefractive soliton as afunction of r and m.

Grandpierre et al. Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. B 59

tan u 5x

z5

v

A2ne~r33uE0u!1/2. (20)

To illustrate our results, let us consider as a first ex-ample a gray soliton of grayness m 5 0.1 and backgroundintensity ratio r 5 40. Let the SBN crystal be 4.5 mmwide, r33 5 280 3 10212 m/V, ne 5 2.33, and let the ap-plied voltage be 2200 V. The wavelength of the laserbeam is 488 nm. Under these conditions one finds fromEq. (19) that m 5 0.84 and, from Eq. (18), v 5 0.56. TheFWHM of the soliton spatial width can be deduced fromFig. 3 and in this case is Dh ' 1.6 or, in real units, 9.2

Fig. 4. Phase jump of a gray soliton as a function of r and m.

Fig. 5. (a) Normalized intensity and (b) phase profile of a grayphotorefractive soliton propagating in a SBN crystal when m5 0.1 and r 5 40.

mm. The angle of propagation is calculated to be 3.24mrad, and after 1 cm of propagation the gray notch of thebeam should be displaced by 32.4 mm. These analyticalresults are in agreement with numerical simulations,shown in Fig. 5(a), carried out by a beam propagationmethod. Figure 5(b) depicts the phase profile at each dis-tance, where for demonstration we removed the phaseterm introduced by (m 1 v2/2)j.

4. SOLITON Y-SPLITTING PROCESSAs previously discussed, under reverse bias, the solitonY-splitting process can occur from an even field depres-sion of a constant phase. Under appropriate initial con-ditions, such depressions split into two symmetric graysolitons, which propagate in opposite directions. In factthis Y splitting has been observed experimentally.25,27

In what follows, we provide an approximate method thatcan semianalytically predict the properties and character-istics of the two gray photorefractive solitons generatedduring this Y-splitting soliton process.

Our approach is based on the total Hamiltonian H ofthe dynamical system defined by Eq. (9), which, as weknow, remains invariant during propagation. If we nowassume that the input is such that it eventually generatesonly two gray solitons (with very little radiation, if any),then the input Hamiltonian energy of the system will beequally divided between these two symmetric states.The reason we choose the Hamiltonian over the other twoconserved quantities P, N is because the evolution equa-tion itself [i.e., Eq. (3)] can be variationally derived fromH. Therefore, to use this procedure, we must first findthe static Hamiltonian Hs of a gray photorefractive soli-ton as a function of m, r. This Hamiltonian Hs can beobtained by substitution of Eq. (13) into Eq. (9), in whichcase one finds:

Hs 5 E2`

`

dsH 2r~ y2 2 1 ! 1 2~1 1 r!lnF ~1 1 r!

1 1 ry2G J ,

(21)

where y(s) is the static field profile as obtained from Eq.(14). As a result, the Hamiltonian Hs is expected to de-

Fig. 6. Soliton Hamiltonian Hs /r as a function of r and m.

60 J. Opt. Soc. Am. B/Vol. 18, No. 1 /January 2001 Grandpierre et al.

pend on both r,m. Figure 6 depicts the normalizedHamiltonian Hs /r of a photorefractive soliton [definedfrom Eq. (21)] as a function of both r and m.

Given an initially even field depression, one can obtainthe total Hamiltonian Hi right at the input through Eq.(9). Since this Hamiltonian (or energy) of the system isconserved during evolution and is expected to be equallydivided between the two gray photorefractive solitons,one can then determine the Hamiltonian of each compo-nent through Hs 5 Hi/2. Given the intensity ratio r in-volved in this photorefractive interaction, and since wenow know Hs , the grayness m of each gray soliton can inturn be determined from Fig. 6. From here the normal-ized velocity v and the spatial FWHM Dh can then be es-timated with the help of Figs. 2 and 3, respectively. Fi-nally, the actual Y-splitting angle can then be computedin radians through Eq. (20). Thus, given the initial fielddepression, all the relevant parameters involved in thisY-junction splitting process can be determined semiana-lytically from Figs. 2, 3, and 6. Conversely, one can solvethe problem backwards. In other words, given the out-put data (the two gray solitons), one can determine the in-put depression and applied voltage required to lead tothis interaction.

In the examples to follow, we consider even field de-pressions that are described by the analytical formU(x, z 5 0) 5 r1/2@1 2 esech2(1.76x/x0)#1/2, where x0 isthe spatial FWHM of the input, e is associated with itsgrayness, and r is again the background intensity ratio.In all our examples, e 5 0.99; that is, the input depres-sion is almost black.

As a first example, let us consider soliton Y splitting ina SBN photorefractive crystal. The characteristics of thecrystal are the same as those given in Section 3. Againl0 5 488 nm and the applied voltage is 2200 V. The pa-rameters of the even field depression at the input are r5 40 and x0 5 18 mm. From these input data and Eq.(9) one finds that at z 5 0 the normalized input Hamil-tonian associated with the system is Hi /r ' 4.054.Since the intensity ratio remains unchanged duringpropagation, the two gray solitons should also have r5 40. Furthermore, by assuming that the input‘‘energy’’ is equally divided between the two components,the value of the Hamiltonian of each gray soliton isHs /r 5 Hi/2r ' 2.027. From Figs. 2, 3, and 6 one findsthat the soliton grayness m 5 0.11, the normalized veloc-ity v 5 0.57, and the actual intensity FWHM is 9.2 mm.From Eq. (20) the splitting angle is then found to beu 5 3.3 mrad. We now check the accuracy of these semi-analytical results by numerically solving Eq. (3) underthe same initial conditions. Figure 7(a) shows the opticalintensity during this Y-splitting process over a distance of2.5 cm in this SBN crystal. The associated phase profileof these two gray solitons is also shown in Fig. 7(b).From these simulations we find that the intensity FWHMof each gray soliton is 9.3 mm, m is 0.12, and the splittingangle is asymptotically u ' 3.3 mrad. These results arein excellent agreement with the predictions of our semi-analytical approach.

As a second example, let us consider the same problem(the parameters of the problem are identical to those inthe first example), but now the intensity FWHM of the in-

Fig. 7. (a) Normalized intensity and (b) corresponding phase profiles as obtained from numerical simulations of the soliton Y-splittingfor r 5 40 and when the spatial FWHM of the even field depression is x0 5 18 mm. (c) Intensity and (d) phase profiles duringY-splitting in a SBN crystal when r 5 40 and the initial FWHM of the even field depression is x0 5 14 mm.

Grandpierre et al. Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. B 61

put depression is 14 mm. In this case Hi /r 5 3.34. Fol-lowing the same procedure as before, one finds thatHs /r 5 Hi/2r ' 1.67 and hence m 5 0.22, v 5 0.7, andu 5 4.1 mrad. The actual spatial FWHM of the two graysolitons is also found to be 11.1 mm. Figures 7(c) and 7(d)show the intensity and the phase profile, respectively, ofthis interaction as obtained from numerical simulations.From these numerical results we obtain m 5 0.25 and u5 4.3 mrad, and the intensity FWHM of these states is12 mm. Note that in this second example the error be-tween the numerical and the semianalytical approacheshas somewhat increased. The reason for such a discrep-ancy is related to the presence of radiative dispersivewaves generated during the soliton splitting, as is clearlyseen in Fig. 7(c). As a result, some part of the inputHamiltonian goes into these radiation components, andthus the condition Hs 5 Hi/2 is not exactly observed.

5. STABILITY OF GRAYPHOTOREFRACTIVE SOLITONSIt has been analytically and numerically demonstratedthat dark or gray solitons described by the generalizednonlinear Schrodinger equations can be unstable1,35,36

even though the nonlinearity is of the defocusing type.In these studies the criterion for dark soliton stability wasfound in terms of the derivative dP/dv, where P is therenormalized soliton momentum defined in Eq. (8) and vis the soliton normalized transverse velocity. More spe-cifically, it was shown that when dP/dv , 0 a gray soli-ton is stable, whereas in the opposite case it isunstable.1,35,36 In this section, we will employ this ana-lytical criterion to determine whether gray photorefrac-tive screening solitons are stable.

By substituting Eq. (13) into Eq. (8), one can find therenormalized momentum Ps for a gray photorefractivesoliton:

Ps~v ! 5 rvE2`

`

dsFy~s ! 21

y~s !G2

. (22)

It is evident from Eq. (22) that the renormalized momen-tum is a function of both r and m. This dependencecomes from the product rv [through Eqs. (18) and (19)]and from the normalized field profile y(s), which implic-itly depends on this pair of variables [see Eq. (14)]. Fig-ure 8(a) depicts the renormalized static momentum Ps /rof these gray photorefractive solitons for various values ofthe intensity ratio r and grayness m. On the other hand,Fig. 8(b) shows the renormalized momentum Ps /r as afunction of velocity v for different values of r. As is clearfrom Fig. 8(b), for r & 40 the derivative dPs /dv is alwaysnegative. However, for larger values of r the curve Ps(v)attains a maximum. Thus photorefractive dark solitonsfor intensity ratios r > 40 exhibit unstable regions forrelatively small values of velocity v. In the unstable re-gions the same (Ps , r) can correspond to different valuesof v, which implies the existence of multivalued or multi-stable solutions.35,36 It is interesting to note that in theregime examined (0.01 & r & 1000) the grayness param-eter m of these unstable solutions happens to be rathersmall, i.e., they are almost black. For example, when r

5 40 and v 5 0.14, m 5 3.8 3 1023. On the otherhand, when r 5 100 and v 5 0.22, the grayness is m5 8 3 1023.

Starting from Eq. (8), it is also straightforward to showthat in the region v → 0 (which corresponds to black soli-tons) the curves Ps /r → p for all values of the saturationparameter r.

This instability of gray photorefractive solitons israther surprising, especially if one considers the (1 1 r)factor in Eq. (3), which tends to enhance the availablenonlinearity in the system.

6. CONCLUSIONSWe have provided a comprehensive theoretical study ofgray spatial optical solitons in biased photorefractivecrystals. The properties of these gray solitons, such astheir transverse velocity, spatial width, and phase profile,have been obtained as functions of their normalized in-tensity and degree of grayness. In a certain range of pa-rameters, we have found that their total phase shift ex-ceeds p, i.e., they become ‘‘darker than black’’ solitons.Moreover, by employing the soliton stability criterionbased on the renormalized momentum, we have foundthat these gray photorefractive spatial solitons can be un-stable for r * 40. The soliton Y-splitting process arisingfrom an initially even field depression has been also semi-analytically described by means of a Hamiltonian tech-nique.

Fig. 8. (a) Renormalized momentum Ps /r of a dark or gray soli-ton as a function of r and m, (b) Ps /r as a function of v for vari-ous values of r.

62 J. Opt. Soc. Am. B/Vol. 18, No. 1 /January 2001 Grandpierre et al.

APPENDIX AThe approximation E0 5 V/W used in our analysis holdswhen the crystal width W is much larger than the solitonintensity FWHM. The correction factor f associated withthis expression can be found for gray spatial solitons by amethod similar to that used in Refs. 16 and 17. In thiscase we write V 5 WuE0u 1 f(uE0u)1/2, from which onefinds that

uE0u 5 S 2f 1 Af 2 1 4WuVu

2WD 2

. (A1)

It can then be shown that V and f are given by

uVu 5 2uE0u

aF E

0

s~1 1 r!ds

~1 1 uUu2!1 S aW

22 s D G , (A2)

where

f 5 2uE0u1/2

aE

0

sF ~1 1 r!ds

~1 1 uUu2!2 sG

5 2uE0u1/2

aE

Am

0.999F ~1 1 r!dy

~1 1 ry2!y82

dy

y8G , (A3)

a 5 (k02ne

4r33uE0u/2)1/2, and s is the half-spatial extentof these solitary waves for which y2( s) 5 0.999. The cor-rection factor f can be further evaluated in terms of r andm:

f 5~2uE0u!1/2

aE

Am

0.999

dyS 1 1 r

1 1 ry22 1 D

3 Fm~ y2 2 m ! 1 ~m 2 1 !S 1

y22

1

m D1 S 1 1 r

rD lnS 1 1 rm

1 1 ry2D G21/2

. (A4)

Figure 9 shows f (which is measured in units ofV1/2 m1/2) for various values of m and r, where the param-eters used are identical to those employed to obtain Fig. 5,

Fig. 9. Correction factor f for gray photorefractive solitons as afunction of r and m.

i.e., uE0u 5 4.44 3 104 V/m, a 5 1.74 3 105 m21. Thiscorrection factor is negligible when W is big enough, thatis, for f 2/(uVuW) ! 1 (Ref. 17), or f ! 1 V1/2 m1/2. This isthe case in the example depicted in Fig. 9.

ACKNOWLEDGMENTSThis work was supported by National Science Foundation(NSF), U.S. Air Force Office of Scientific Research, U.S.Army Research Office, U.S. Army Research Office Multi-disciplinary University Research Initiative, and Austra-lian Photonics Cooperative Research Center. The re-search of M. Segev was also supported by the fund for thePromotion of Research at the Technion.

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