Page 1

670

OPTICS LETTERS / Vol. 23, No. 9 / May 1, 1998

Stabilization of dark and vortex parametric spatial solitons

Tristram J. Alexander

Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University,

Canberra, ACT 0200, Australia

Alexander V. Buryak

School of Mathematics and Statistics, Australian Defence Force Academy, Canberra, ACT 2600, Australia, and Optical Sciences Centre,

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

Yuri S. Kivshar

Australian Photonics Cooperative Research Centre, Research School of Physical Sciences and Engineering, Optical Sciences Centre,

Australian National University, Canberra, ACT 0200, Australia

Received December 22, 1997

We demonstrate that a weak defocusing Kerr effect in an optical medium with predominantly quadratic [or

x?2?] nonlinear response can eliminate the parametric modulational instability of plane waves, leading to the

existence of stable two-wave dark and vortex spatial solitons.

OCIS codes:

190.7070, 190.5530, 190.3270.

1998 Optical Society of America

Dark spatial optical solitons, or self-trapped station-

ary dark notches existing on an infinite plane-wave

background or embedded in wide diffracting beams,

have been investigated for a number of years (see, e.g.,

the recent review in Ref. 1).

is due to potential applications of spatial solitons in

photonics as steerable and reconfigurable self-induced

waveguides.Up to now, the experiments were limited

by materials with relatively weak nonlinear response,

and therefore successful demonstrations of dark soli-

tons and the waveguides that they induce was achieved

for just a few optical materials.2,3

in recent years remarkable progress was made in

the field of so-called cascaded nonlinearities of opti-

cal materials with a quadratic [or x?2?] nonlinear re-

sponse.4

In particular, it was shown that both bright

and dark parametric solitary waves can exist as mutu-

ally trapped two-color beams of the fundamental har-

monic (FH) and the second harmonic (SH).5

straightforward extension of the concept of spatial

dark solitons to the case of quadratic optical materials

fails, because all finite-amplitude plane waves (PW’s)

in a pure x?2?medium suffer from parametric modu-

lational instability.6

As a result, no stable spatial

dark solitons have been found to exist in pure qua-

dratic media.6,7

In this Letter we suggest a simple physical mecha-

nism leading to stable dark solitons in optical me-

dia with quadratic nonlinearities.

the stabilizing action of the defocusing cubic (or Kerr)

nonlinearity, which can eliminate parametric modula-

tional instability in two-wave mixing.

the instability, in turn, allows the existence of dark

and vortex spatial solitons stabilized by the presence

of a weak cubic nonlinearity in an optical medium with

a predominantly quadratic nonlinear response.

cubic nonlinearity may be inherent, i.e., due to the

next-order nonlinear contribution of a noncentrosym-

metric optical material,8or induced, e.g., generated by

an incoherent coupling of FH and SH beams with other

Interest in this research

On the other hand,

However,

First, we reveal

Suppression of

This

modes or higher-order spatial frequencies in quasi-

phase-matched materials.9

Physical equations that describe the interaction of

the FH ?v1? v? and the SH ?v2? 2v? in a medium

with both x?2?and x?3?nonlinearities are well known

and can be found in earlier papers.8

waveguide geometry, we measure the transverse coor-

dinate in the units of the beam size R0and the propa-

gation coordinate in the units of the diffraction length

Rd ? 2R02k1, where k1? k1?v? is the wave number

of the FH. Then, making the scaling transformations

similar to those in Ref. 8, we obtain the system of two

coupled equations for the normalized envelopes of the

FH ?u? and the SH ?w? in the form

Assuming a slab

i≠u

≠z1 r=?2u 2 u 1 wu?1 x

µ1

4juj21 rjwj2

∂

u ? 0,

2i≠w

≠z1 r=?2w 2 aw 11

µ

2u21

x

4jwj21 rjuj2

∂

w ? 0,

(1)

where r ? sign?b?, a ? ?4b 1 2DkRd??b, b is the

nonlinear contribution to the propagation constant

of the fundamental wave, Dk ? ?2k1 2 k2? is the

wave-vector mismatch between the harmonics, r is

the cross-phase modulation coefficient, the coefficients

x?2?and x?3?are proportional to the relevant ele-

ments of the corresponding nonlinear susceptibility

tensors, and, for the ?1 1 1?-dimensional geometry

case, we should take =?2? ≠2?≠x2.

ter x ? 3bc2x?3???16pv12R02?x?2??2? characterizes the

relative contribution of the cubic nonlinearity, and its

sign is defined by that of the product bx?3?.

we consider both r ? 11 and r ? 21, so Eqs. (1) have

only two continuous parameters, a and x, except for r.

We keep r unfixed to present the exact soliton solution

The parame-

Below,

0146-9592/98/090670-03$15.00/0

1998 Optical Society of America

Page 2

May 1, 1998 / Vol. 23, No. 9 / OPTICS LETTERS

671

[Eqs. (3), below] but set it as r ? 2 in the rest of the

analysis.

We can find solutions of Eqs. (1) for PW’s by solving

the algebraic equations 12xwp31 12wp21 wp?a 2 8 1

2?x? 2 2?x ? 0 and up2? 4?1 2 wp??x 2 8wp2for real

upand wp.There exist as many as three branches of

PW’s for which both amplitudes upand wpare nonzero.

Performing standard parametric modulational insta-

bility analysis (see, e.g., Refs. 6 and 8) for each of the

three branches of PW solutions at r ? 61, we find that

there exists only one branch of modulationally stable

PW solutions.The parameter domains in which such

a solution exists are presented in Fig. 1.

sign?rx? ? signx?3?, and thus the modulationally stable

PW’s exist only for x?3?, 0.

Importantly, the amplitudes of the modulationally

stable PW’s diverge in the limit x ! 0, as is clearly

seen in Fig. 2, so the stable branches of the PW’s exist

exclusively because of the mutual action of quadratic

and cubic nonlinearities.Other PW’s are modulation-

ally unstable in the whole domain of their existence,

and they are not presented in Figs. 1 and 2.

tional stability of PW’s in the limit of large negative x?3?

[e.g., for r ? 21 and x . 0; see Figs. 1(b) and 2(b)] is

not surprising because stable dark solitons are known

to exist in a defocusing Kerr medium.

interested in the case in which the effective nonlinear-

ity is predominantly quadratic, i.e., jxupj ? jxwpj , ,1.

We found that this condition can be satisfied only for

r ? 11, where modulationally stable PW’s of moderate

amplitudes exist for relatively small values of x [see

Figs. 1(a) and 2(a)].

Modulational stability of PW’s is promising for the

existence of stable dark solitons.

solitons are described by the system of two second-

order ordinary differential equations for the real func-

tions u?x? and w?x?,

Note that

Modula-

Here we are

For our model dark

rd2u

dx22 u 1 wu 1 x

µ1

4u21 2w2

∂

u ? 0,

rd2w

dx22 av 1

with nonvanishing boundary conditions.

these stationary solutions numerically, applying the

relaxation technique.As a result, we find that dark

solitons exist in the whole domain of the modulation-

ally stable PW’s shown in Fig. 1, and characteristic ex-

amples of such solitons are presented in Fig. 3.

It is interesting to notice that at a ? 1??r2x? Eqs. (1)

have the exact solution

µ4

1

2u21 x?4w21 2u2?w ? 0,

(2)

We obtain

u ?

x1

1

rx2

∂1/2

tanh

"

x

µ

2r

22

r

8rx

∂1/2#

,

w ? 21??2rx?,

(3)

which exists for x . 0 ?r ? 21? or 21??4r? , x , 0 ?r ?

11?.For r ? 11 and r ? 2 and close enough to the

value x ? 21?8, we have jxupj ? jxwpj ? 0.2.

As is shown in Fig. 4, on the parameter plane ?x, a?

the exact solution (3) separates two types of dark soli-

ton:

solitons with a dip-shaped SH; see Figs. 3(c) and 3(d)]

and the other for which the SH component has a maxi-

mum [i.e., dark solitons with a hump-shaped SH; see

Fig. 3(b)].

In spite of the fact that the dark solitons that

we found here exist on a modulationally stable PW

background, they can become inherently unstable,

displaying so-called drift instability.10

tain the stability criterion for dark solitons described

by Eqs. (1) by generalizing the formalism of Ref. 10,

which, however, requires knowledge of more-general

gray solitons, i.e., dark solitons propagating under a

one for which the SH has a minimum [i.e., dark

We can ob-

Fig. 1.

PW’s:

Existence domains for the modulationally stable

(a) r ? 11, (b) r ? 21.

Fig. 2.

versus x at a ? 22.0.

respectively.

Amplitudes of the modulationally stable PW’s

(a), (b), Figs. 1(a) and 1(b),

The limit x ! 0 is singular.

Fig. 3.

11:

x ? 20.1, (c) a ? 21.0 and x ? 20.1, (d) a ? 4.0 and

x ? 20.05.

Examples of dark solitons of Eqs. (1) at r ?

(a) a ? 22.5 and x ? 20.1, (b) a ? 24.0 and

Page 3

672

OPTICS LETTERS / Vol. 23, No. 9 / May 1, 1998

Fig. 4.

r ? 11.

solution [Eq. (3)].

dark solitons presented in Figs. 3(a), 3(b), 3(c), and 3(d),

respectively.

Existence domains for dark solitons of Eqs. (1) at

The curve x ? 1??4a? is defined by the exact

Points A, B, C, and D correspond to the

Fig. 5.

x ? 20.1, a ? 22.5, and r ? 11.

for the FH. (b) Radial dependence for the FH and

the SH.

Two-wave single-chargedvortex

(a) Intensity profile

solitonfor

certain angle to the background wave.

of gray soliton families is still an open problem even

for the case of pure x?2?nonlinearity.

confirmed the soliton stability directly by modeling

the propagation of dark solitons numerically for suf-

ficiently large distances.

The analysis presented above can be readily ex-

tended to the case of higher dimensions, i.e., for dark

solitons of circular symmetry in a bulk (vortex soli-

tons) associated with a phase singularity embedded

in a background wave (see, e.g., Ref. 1).

structure of the vortex solitons, we look for radially

symmetric solutions of the ?2 1 1?-dimensional version

of Eqs. (1) with =?2? ≠2?≠x21 ≠2?≠y2, in the form

u?R, w? ? U?R?exp?imw?, w?R, w? ? W?R?exp?2imw?,

where R ? ?x21 y2?1/2, w ? tan21?x?y?, and m is

an integer number (the vortex charge).

the lowest possible charge ?m ? 61? corresponds to a

2p phase twist of the FH and a 4p phase twist of

the SH. Solutions of the corresponding equations for

the amplitudes U?R? and W?R? satisfy the conditions

U?0? ? W?0? ? 0 and dU?dR ? dW?dR ? 0 at R ! `.

Figure 5 shows an example of a stable two-wave vortex

soliton of the first order, jmj ? 1.

for the parameters x and a shown by point A in Fig. 4.

We expect that stable dark and vortex solitons in

quadratic media can be observed in typical upconver-

sion experiments when a high-intensity beam under-

goes frequency doubling simultaneously with creation

of a phase singularity produced by a phase mask at

the input, similar to recent experiments performed at

moderate powers.11

At higher input powers, the ap-

This finding

However, we

To find the

A vortex of

The soliton exists

propriately chosen nonzero phase matching will lead to

the formation of a dark (or vortex) soliton stabilized by

the effect of incoherent coupling between the harmon-

ics owing to the cubic nonlinearity.

note that optical materials used in these experiments

should demonstrate the defocusing Kerr effect for both

harmonics. The most-likely candidates for such ma-

terials are poled optical polymers (see, e.g., Ref. 12),

which often have negative x?3?in a large range of op-

tical frequencies.The minimum light intensity that

is required for generation of a stable dark soliton in a

predominantly quadratic medium can be estimated as

Imin? ?x?2??x?3??23 109, where x?2?and x?3?should be

inserted in picometers per volt and picometers squared

per volt squared, respectively, whereas Iminis given in

megawatts per square centimeter.

In conclusion, we have demonstrated the existence

of modulationally stable plane waves for the process of

two-wave mixing in a parametric medium with both

a quadratic and a cubic nonlinear response and found

the families of dark and vortex spatial solitons with

a nonvanishing background field.

that these dark solitons can exist in the region of

parameters that correspond to a predominantly x?2?

medium and are therefore stabilized by a weak cubic

nonlinearity. This result gives us what we believe

to be the first example of stable spatial dark solitons

supported by quadratic nonlinearities.

It is important to

We have shown

The authors thank Ole Bang and Stefano Trillo

for useful discussions.A. V. Buryak acknowledges

support from the Australian Research Council.

References

1. Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 288,

81 (1998).

2. B. Luther-Davies and X. Yang, Opt. Lett. 17, 496

(1992).

3. M. Shih, Z. Chen, M. Mitchell, M. Segev, H. Lee, R. S.

Feigelson, and J. P. Wilde, J. Opt. Soc. Am. B 14, 3091

(1997).

4. See, e.g., G. I. Stegeman, D. J. Hagan, and L. Torner,

Opt. Quantum Electron. 28, 1691 (1996).

5. For a classfication of bright and dark solitary waves in

diffractive x?2?media, see, e.g., A. V. Buryak and Yu. S.

Kivshar, Phys. Lett. A 197, 407 (1995).

6. S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).

7. A. V. Buryak and Yu. S. Kivshar, Phys. Rev. A 51, R41

(1995); Opt. Lett. 20, 834 (1995).

8. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Opt. Lett.

20, 1961 (1995); S. Trillo, A. V. Buryak, and Yu. S.

Kivshar, Opt. Commun. 122, 200 (1996); O. Bang,

J. Opt. Soc. Am. B 14, 51 (1997).

9. C. B. Clausen, O. Bang, and Yu. S. Kivshar, Phys. Rev.

Lett. 78, 4749 (1997).

10. D. E. Pelinovsky, Yu. S. Kivshar, and V. V. Afanasjev,

Phys. Rev. E 54, 2015 (1996); Yu. S. Kivshar and V. V.

Afanasjev, Opt. Lett. 21, 1135 (1996).

11. K. Dholakia, N. B. Simpson, M. J. Padgett, and L.

Allen, Phys. Rev. A 54, R3742 (1996); A. Berzanskis,

A. Matijosius, A. Piskarskas, V. Smilgevicius, and A.

Stabinis, Opt. Commun. 140, 273 (1997).

12. C. Fiorini, F. Charra, P. Raimond, A. Lorin, and J-M.

Nunzi, Opt. Lett. 22, 1846 (1997); H. Nakayama, O.

Sugihara, and N. Okamoto, Opt. Lett. 22, 1541 (1997).