Three-wave polariton solitons of new type in polar semiconductors

003%1098/91$3.00+.00 Pergamon Press plc

Solid State Communications, Vol. 78, No. 9, pp. 787-792, 1991. Printed in Great Britain.

THREE-WAVE POLARITON SOLITONS OF NEW TYPE IN POLAR SEMICONDUCTORS A. L. Ivanov Department Leninskii (Received

of

and G. S. Vygovskii

Physics, Moscow State University, Gory, Moscow, GSP 119899, USSR 18 March

1991

by L. V.Keldysh)

The formation of coupled three-wave solitons of new type under resonant interaction of two polariton waves by means of LO-phonons in polar semiconductors is first discussed. The consistent consideration of polariton effects enables to give solutions by their classification of the soliton dispersion. The possibility of the formation of quasi-phonon solitons with unusual properties is first reported.

The formation of the coupled three-wave solitons has been studied intensively in the 1960’s mainly in the nonlinear optics since framework of phenomenological approach [I,21 and at present remains in the focus of considerable research [3-91. The solitons in stimulated Raman scattering (SRS) were first observed in 1983 in COz-pumped para-Hz [31. The main purpose of this

a2 at2

=

C’

a2

;

& - w

,&

L P A/=

;

Here M denotes the translational =x Em- background dielectric constant

paper is to attract attention to a novel subject of interestthe problem of the three-wave solitons formation in the Raman interaction of two excitonic polariton waves in direct-gap semiconductors where the large optical nonlinearitles enable to utilize the electromagnetic pulses of moderate intensity. The dynamic behavior of exciton-photon-LOphonon system in a polar semiconductor in the presence of polariton pump wave of the given intensity has been investigated recently in [IO121. In this paper we are mostly interested in the nonlinear propagation of two comparable in intensity coherent polariton waves under their strong resonant interaction via LO-phonons. The corresponding set of macroscopic equations for the positive-frequency components of the electexcitonlc romagnetic fields E(r, t 1, ecr, t I, polarizations P(r, t). P(r, t) and LO-Dhonon scadeiived from potential i(r,.t) can be iar consistent microscopic approach: 4~ E=-_-__P

“: g

p-

crystal

wy- exciton

density, energy,

RO-

of

LO-phonon

7 and a” -

of exciton and LO-phonon, (I) contain two important

exciton

crystal, frequency,

inverse

respectively. constants:

mass,

lifetimes The Eqs. the dimen-

sionless parameter i characterizing the strength of the exciton-photon interaction and the parameter L [Ill for Frohlich exciton-LO-phonon interaction in polar semiconductors. For simplicity in the set (1) we neglect the crystal anisotropy and consider only the case of collinear polarizations of interacting polariton waves: e II e.. In what follows the real dissipative P

k

processes

described

by

the

corresponding

constants 7 and ra are also neglected. Let us discuss the physical origin of the set (I). Eqs. (la), (lb) and (Id), (If) describe the propagation of the polariton waves interacting via phonon field. Eqs. (la1 and (Id1 are usual Maxwell wave equations for the electromagof the first and the second netic component polariton wave, respectively. Eqs. (lb) and (If) describe the propagation of the excitonic polawhose linear source is deterr izat ion waves, mined by the excitons creation by the corresponding electromagnetic wave, while the nonlinear source is associated with the coherent Raman scattering of excitons from one polariton wave into another. And finally Eq. (1~) characterizes the behavior of the coherent longitudinal phonon field arising from the nonlinear Raman interaction of the two intensive polariton waves. LO-phonon scalar potential @(r, t) is related to the corresponding lattice displacement field u(r,t) = V O(r, t) . u(r.t) by: The further analysis of the set (1) will be devoted to the investigation of the possibility

(la)

at2

(lb)

(lc)

(Id)

787

THREE-WAVEPOLARITON SOLITONS OF NEW TYPE

788

of formation in semiconductors of the three coupled nonlinear waves, two polariton and one LO-phonon, dispersive propagating without spreading with the common group velocity u . For 8 simplicity we conf fne ourselves to the onedimensional wave interaction taking as a model a semiconductor crystal CdS: pllkll
rather than electromagnetic fields are used as variables in the set (21. This enables to consider the most interesting case when the carrier frequency w of the first polariton wave lies close to the anti-Stokes resonance wk+ R of the 0 second wave or to the exciton resonance w and t when the usual procedure of expansion x = x(w,p, only

)p

0 the

= -ia

$ + id

p’

p

;

(2bl

dt

where the the

‘c=

-

ipa1PI”

- the retarded

t-c/u

envelope following Zdt&

P + iH2p Z’

,

time,

T(r)

denotes Here

if the corresponding field. definitions are introduced: c

L (p+k)

c*= pf- $

P=1

lLl;

;

C

si”s

(p+kh&

L

B,= SU 1

(Zc)

!

D

(p+k)o.

+ 2a

;

(31

si=

2

w+1

t

PI hot u

8

M

ex

I

4s; + T CC

W2” tl 1

(

2+-

blwl C

; 1

i

hw a,=

w;-

w;+

; b,= 2

p: 2 M

ex

B

L-l:- (w-wk12

L

a= 2(w-wr 1 for

i = 1,2

d = w,&

lowest

and wl= w, w2= y,

Unlike the optics nonlinear

p,=

-p,

p,=

k.

approach traditional for the relevant polarizations the

power

order

series

and

nonlinear

treating

susceptibiThe set of to account of nonlinear

of the susceptibilities x (“) in the framework initial microscopic model. At the same time the proposed form of the Eqs. (21 naturally deals with the fact that polariton waves interact just via polarization components, i.e., exciton components. Note that in the set (21 wave vector and frequency matching condition is supposed to be satisfied. Introducing the real amplitudes a, 4. 5 and and I) for the positive-frequency phases (P. x, soliton envelopes p = a eicp ;

a’

=

4’

c -8,

F = 4 eix

the

final

P, 4 4 sin a ‘& sin

Ql = -day

;

; = /& ei*

set:

Q

;

Q

;

sinQ ; p,cC2)a + 8, 4 4 cos Q p,a2)v+ i3,a 5 cos Q

4 a/=

-(q2+

‘t q=

- a++dagcosQ

(41

: ;

,

where Q = (p-x-$ - the phase-matching angle for interacting polarization waves. The first three amplitude equations from set (41 have three integrals of motion of Manlay-Rowe type, two of them are independent: Baa2 + B,cc2 =cl

;

-dq2 + P2q2 =c2

;

da2 + ,!j,g2 = c3

;

(51

The analysis of the integrals motion (51 enables to carry out the complete classification of all possible soliton solutions by the signs of parad and values of constants C1. In meters 6 H 1’ 2’ the nonlinear optics the three-wave solltons are defined as such three stationary waves, at least two of them are solitary, i.e., their amplitudes vanish at ‘c + i: m . In fact such a definition corresponds to the case when two solitary coupled nonlinear pulses form a stationary nonlinear perturbation in the cw background of the third wave, which may be referred to as a pump wave. The described scheme of three-wave sollton formation leads to the requirement of vanishing of one of the constant C, in the integrals of motion

G 2(W_Wkl

a

lities, i.e. ~(~1 and ~(~1, fails. polarization equations (21 allows explicitly for the entire series

a (p'=-(ql+

d?’ = -iq2+ dt

into

I21

we obtain

-

Vol.78,No.9

(51.

The

case

C,= 0,

when

the

coherent

wave of LO-polarization appears to be a pump is hardly to be realized experimentally wave, and so will not be analyzed here. The condition C2= 0 for the pump polariton wave with the

Vol. 78. No. 9 carrier Stokes

frequency w in the vicinity resonance wk+ Roof the second

of antipolariton

pulse deals with the formal construction of stationary nonlinear solutions of the set (41 unstable concerning its possible evolution. Such instability is determined by the processes of spontaneous Raman decay of the pump wave p into LO-phonon wave p-k and Stokes component k. It is known [3-91 that the evolutlon of this instabllity leads to the stimulated Raman scattering resulting in the generation of coupled threewave solitons with structure however corresponding to the case of the pump wave at the Stokes Thus in the most interesting case frequency. which will be analyzed further Cs= 0, i.e., the second polariton wave acts as a pump wave. Then for the selected condition Cs= 0 one can find the following expressions nary envelope amplitudes of type determined by the sign of parameter corresponding angle Q :

polarizations 4 (l-

and

for statioessentially pa, for the

phase-matching

1 l/2

B2/41

4’4 o

[

1il-zzBch1;~2’18

; (6)

-2 1 l-

B2/4

where t = t

, B and

K

are defined

by:

2 P, 4:

(8) ; is

an

K = - 4,&y-

amplitude

The case

of

S,
; polarization upper

(P‘Ir_tkrn

=

*’ ltj+oo= .Y’IT& = 0

’

of

the

(10)

from the latter three Eqs. of set (4) with the help of the integrals of motion (51 we obtain two additional independent relations between the parameters of the problem:

Ql +

;

a = - p,4;

(lla)

q=o

(lib)

2

which actually are the dispersion equations for the carrier frequencies and wave vectors of the first and the second polariton wave, respectively. Thus three relations (9). (lla) and (lib) establish the relationship between the initial parameters of the problem. We consider reasonable from a physical point of view to select the following parameters as the independent ones: the carrier frequencies w and wk of polariton pulses determined by the external sources, the intensity of the pump wave, directly related to the quantity 4,, and the duration T* of the soliton pulses. The relation (lib) represents an usual polariton dispersion law for the pump wave reflecting its free propagation in the absence of two solitary pulses , i.e.. at corresponding T + * m. The equations (91 and (llal represent the complete set of algebraic equations for the soliton velocity u = u (w,w k, 10,tsl and the car8 s rier wave vector p = p(w,w,, lo,~a) of the solipolariton wave. Then we obtain the disperrelation for the first polariton pulse in of the algebraic equation of the 9-th order respect to the wave vector p. The typical dispersion p = p(w) for the given values of parameters ok, Ioand 7= Is pre-

8

B = - .&

vanishing phase modulation Assuming soliton solutions at r + f m

tary sion form with

J4+k2+2mE sh 2;

tg Q =

and 4 0 r++z:m.

789

THREE-WAVE POLARITON SOLITONS OF NEW TYPE

sign

sented in Fig. 1. At any of the plotted dispersion branches the soliton velocity u has a deB finite sign determining the polariton waves interaction geometry for the excitation of any

%’ at in

(61)

corresponds to the situation traditionally considered in the theory of three-wave interactions [2-51 when two nonlinear solitary waves ‘burn’ a dark soliton in the pump wave, i.e., a dip in the pump intensity profile. In the case p,>O (the lower sign in (6)) the solutions (6) for the shapes of stationary nonlinear pulses characterize the situation when the interaction of two solitary waves with the pump leads to the formation of a spike in the intensity profile of pump wave. Note that the parameter K differs in other words, the odd-order from zero or, prevent the nonlinear susceptibilities x ‘2”“) formation of the explosive instability. It is that regardless of the type of lmportant solutions (6)-(71 the characteristic duration of soliton pulses TV is given by:

WAVE VECTOR (lO’cm_‘) Fig.1 for

The typical ok= 2.513 eV.

dispersion

p = p (01 plotted

IO= 500 MU/cm2 and T = 15 ps.

The inset illustrates the configuratlo: of polariton waves interaction. The parameters are taken for MS: ut= 2.552 eV, RO= 38 meV.

790

THREE-WAVE

POLARITOiJ SOLITOdS

selected branch. Thus along the dispersion branches 1, 3 and 5 the velocity u < 0 and therefore B their excitation is possible in the forward scattering geometry. The dispersion branches 2 and 4 with soliton velocity IP~>0 correspond, in turn, to the backward scattering geometry. In the certain context the appearance of new dispersion branches for the carrier wave vector of the solitary polariton pulse is associated with the strong renormalization of the initial LOphonon term and unperturbed polariton dispersion analogous to the recently investigated in [lo121 phonoriton spectra renormalization in the presence of the stationary polariton pump wave. The analysis of the behavior of dispersion curves l-5 under the variation of intensity I 0' in particular at I + 0, enables to perform the 0 following classification of the solutions. Namely one can relate the dispersion branches 1 and 2 to the corresponding soliton solutions with polariton features that will be denoted as quasi-polariton solitons. The solitons with dispersion branches 3-5 may be said to be 'topological fragments' from the unperturbed LO-phonon term. The nonlinear polariton pulses with the specified dispersion laws corresponding to the second solution type are denoted as quasi-phonon solitons. The dispersion branches 4-5 represent closed curves which resemble spectral 'droplets'. They disappear with diminishing pump wave intensity and will not be analyzed here. The existence of the branch 3 is connected with the invariant way of interlocking of the initial unperturbed dispersion curves in their evolution towards the modified spectrum. Giving the general characteristics for the considered dispersion features one should note that the strong modification of initial terms of the interacting waves in the vicinity of the anti-Stokes resonance leads to the breakdown of the relevant analysis by means of the perturbative approach. Indeed, for the analysis of the analogous problem of the SRS-solitons formation one uses the unperturbed or slightly perturbed initial terms as the dispersion relations for soliton pulses. At the best this approach yields the results qualitatively relevant to the first type of the solutions specified above, i.e., quasi-polariton solitons. As regards the second type of the solutions - quasi-phonon solitons , it cannot be studied in this approach at all. It should be noted that this statement is valid also in a more general case when the origin of the exciton-phonon interaction is not Frohlich one. Let us discuss in more detail the features of the quasi-polariton soliton solutions. In accordance with the opposite signs of the parameter pz(o.wk.I ,T*) for the dispersion branches 0 1 and 2 (see Fig.11 the shapes of three-wave soliton pulses differ qualitatively depending on the geometry of interaction - forward- or backscattering- in which they are excited. The results of numerical calculations of the shapes (6) of coupled polariton solitons are presented in Fig.2. Note that the solitary polariton wave may be defined as a giant parametric pulse. The dependence of the soliton velocity u = u (w) at different values of the pump intens s

Vol. 78. No. 9

OF NEW TYPE

5 ,",

s

4-

d L, 2

3-

‘, ,, ::I,: ,I : ,, ,< :: ::

b

TIME (ps)

Fig.2. The shapes of coupled polariton solitons for the quasi-polariton branches 1 (al and 2 (bl plotted for the same group velocity us. Solid lines corresponds to the exact anti-Stokes resonance, dashed - for Iwk+ Ro-Otl= 0.04 meV. Pump intensity IO= 50 MU/cm'. is presented

sity and fixed soliton duration r

in Fig.3 for the forward and Sbackscattering interaction geometry. As shown, the quasipolariton soliton velocity tends to decrease when approaching the anti-Stokes resonance and with increasing pump intensity. For the fixed parameters wk, I and T at frequencies w suffis ciently far f;om the anti-Stokes resonance wk+ R the quasi-polariton soliton propagates at 0 the usual polariton group velocity, or more accurately, at the phonoriton velocity. It should be emphasized that the set (41 derived following the been using has assumptions:

(12)

7

5

I

E

p2- 2 w2 c2

I I > 2

C t

s

k2-

2

c2

E

Pw-_ C2 E

u:

> I

2 I

2w+k C2

P

;

(13a)

c9 k”

; a

(13b)

8

The inequality (121 determines the condition for the validity of the slowly varying envelope approximation. The inequalities (13al and (13b) enable to express explicitly the electromagnetic

Vol. 78, No. 9

THREE-WAVE

POLAFCITON SOLITONS

OF NEW TYPE

-2.0

5

E

"-2.5 3; : -3.0 ki w" -3.5 2 k% -4.0 L -1.0

-0.5

0.0

0.5

1.0

VELOCITY (lO*cm/s)

0.4

WAVE

VECTOR

0.2

0.0

(lOacm_') VELOCITY (105cm/s)

Fig.3. The dependence of the soliton velocity us= 098(wI for quasi-polariton solltons. Pump

Fig.4. The quasi-phonon soliton dispersion branch 3 and corresponding soliton velocity

intensity I = 200 MW/cm2(1) and I =500 MW/cm' 0 0 (21, T = 10 ps. Dashed line corresponds to 8 polarlton group velocity.

tips= as(w). Pump intensity IO= 200 MU/cm2 (11 and

components i and g of polarlton waves via cor-

the abnormal dispersion which can be formally related to the relevant region of the unperturbed phonon dispersion line w = ok+ G . In a 0 sense one can say about an unusual quasi-phonon sollton velocity uB behavior, i.e., it increases

responding polarizations ?,and ?. The validity of the conditions (12)-(13b) depends essentially on which sollton dispersion branch they are being considered. The specified relations are always satisfied for the quasipolarlton dispersion curves in the case of not extremely high pump intensities (IO-:1 GW/cm2). As the intensity I

increases, the inequality

(13b) for the nonlcnear pump wave is first to fall. As regards the quasi-phonon dispersion branch 3, the condition (13bI is never valid for It. To overcome this problem we present below a specific approach for the analysis of the quaslphonon solltons. The failure of the condition (13b) for the quasi-phonon solltons requires unlike the prevlously described approach to account more accurately for the polariton effects for the nonlinear pump wave. In other words, one should treat the Eqs. (Id) and (If) instead of Eq. (2~) to obtain a closed set of four nonlinear dlfferential equations for positive-frequencyenvelopes F, i, ? and .%. Unfortunately, this set is rather complex for analytic analysis and requires a numerical study and will be analyzed in detail elsewhere. Here we shall only mention the main results. First, the conservation of the form of the relations (91, (lla) and (lib) between the lnltlal parameters of the problem for the more general set should be emphasized. This enables to utilize the previously presented classification of the soliton solutions by the types of the dispersion branches for the solitary polariton pulse. The quasi-phonon soliton dispersion branch 3 and corresponding sollton velocity ue= @*(oI at the various values of intensity I

and fixed

sollton duration T. are shown in Fig.:. The main tendency lies in the increasing with the lncreasing intensity of the characteristic region of

IO= 500 IN/cm2 (2). ts= 15 ps.

as the pump intensity grows. Such a behavior differs qualltatlvely from the corresponding dependences u = uss(o,IO)for the previously cons sidered quasi-polarlton solltons as well as for the ones traditionally investigated in the nonlinear optics showing decreasing soliton velocity with increasing intensity IO. The numerical computations of the quasiphonon soliton envelopes behavior show that the stable three-wave solitons of the quasi-phonon exist

type

only

at

I r I',h,

where

-the

rp= I~(w.r&

threshold pump wave intensity. For the considered semiconductor CdS

_:. .___.--_ 0.0

I

0

100

TIM$ps) Fig.5. Evolution of the quasi-phonon soliton envelopes at the exact anti-stokes resonance w = ok+

n

for

pump

intensity

I = 40 MW/cm2

(dashed)lnd IO= 200 MW/cm2 (solid Ilone).

792

THREE-WAVE F'OLARITONSOLITONS OF NEW TYPE

Ith= 100-200 MU/cm2 critically depending on the soliton duration r The appearance of the in8. stability and its evolution under the variation of the pump wave intensity at the exact antiStokes resonance w= wk+ R are shown in Fig.5. 0 As regards the frequency region for the existence of the quasi-phonon solitons, it is located in the vicinity of the anti-Stokes resonance and in fact is related to the abnormal dispersion region of the dispersion branch 3 being analyzed.

Vol. 78, No. 9

In conclusion, the proposed theoretical approach and the main results may be applied also for the analysis of the formation of polariton solitons due to the Raman interaction in molecular crystals and polymers. The latter are of the most interest in the context of the recent experiments reported in [13,141.

The authors would like to thank Prof. L.V.Keldysh and Prof. A.P.Sukhorukov for helpful discussions. Acknowledgements-

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3. 4. 5. 6. 7. s.

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9.