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Combination resonances of vortical type
Volume 129, number I
PHYSICS LETTERS A
2 May 1988
COMBINATION RESONANCES OF VORTICAL TYPE V.E. SHAPIRO Kirensky Institute of Physics, Krasnoyarsk 660036, USSR Received 2 November 1987; accepted for publication 26 February 1988 Communicated by V.M. Agranovich
A vortical mechanism of energy transfer between systems with different frequencies of motion is investigated, which performs the energy conversion due to the combination resonances differing from the conventional concept. General remarks and the simplest considerations of the vortical mechanism display in stimulated Raman scattering are presented. The possibility of a considerable increase of the scattering is pointed to.
1. Introduction In stimulated Raman scattering and many other phenomena of high frequency effects in multidimensional systems one meets with concurrence and interaction of a great number of resonantly combined waves. For the combination resonance of a given order the minimum number of waves, corresponding to the number of quanta in the elementary reactions of energy transfer along the spectrum, plays an important role in the interactions. But not all the basic processes inherent in the resonance energy exchange are reproduced with account of only the minimum number of waves. In fact, only one type of process is reproduced which, for considerations explained below, will be called dissipative. There exists a different type of combination resonance process called vortical, which, in a certain sense, is complementary to the dissipative one [1]. Relying on the results of ref. [1] the present note aims at extending this point. Let, for example, only the first order combination resonances (0 = 0~k+
Q,
~
=
c0k —
Q
be a factor, i.e. waves of frequency
(1) co excite
in a me-
dium oscillations of low frequencies Q and waves of Stokes and anti-Stokes frequencies Wk arise. It is often believed for this case that modelling of the wave reservoir by three nonlinearly coupled waves (oscillators) of frequencies co, Wk and Q re62
produces the main resonance processes, that buildup of the combination oscillations arises due to the resonances (1) for which co> ~0k and those forwhich w < COk cause the opposite effect, suppress the oscillations, and that at least in a qualitative aspect this guide is satisfactory for weakly nonlinear systems. But these assertions need revision for systems in which both high and low frequency forms of motion involved in the interactions are multi-dimensional and have a degenerate spectrum. Under such conditions the mentioned vortical combination processes become significant. They tend to involve a number of triples of resonantly combined waves into the coordinated motion and lead to the trend of the resonance phenomena is quitescheme inexplicable from the position of thewhich three-wave of the analysis. In particular, for weakly nonlinear systems, within the three-wave terms in the interaction energy accounted in the second order perturbation theory, build-up of the combination oscillations becomes possible for all values of w including w< min{wk}. The idea of the combination resonance processes with off-beat character may be suggested from symmetry considerations. Let X= (X 1 (t), X2(t), ...) be the variables charactensing the oscillatory (at low frequencies Q) medium state of different spatial forms and F= (F1, F2, ...) be the combination forces conjugated to X. Per cycle of periodic change in X the forces F perform the work
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 129, number I
PHYSICS LETTERSA
W ~‘F~dXi. Here and further summation over repeated indices is implied. The forces F are functions of the scattered high frequency field. Since this field itself is brought about by the oscillations X and a retardation takes place the forces F are a retarded functional of X. In the general case F can be divided into three parts: F= F~ + F +F d where F’ includes all the reversible part of F, which does not contribute to W. The irreversible forces F v are even and F” are odd with respect to the change of the direction ofthe motion Xinto the opposite one, The irreversible forces F’~changing sign when the motion X inverts its direction are forces like friction and dissipation. When representing the negative friction they stimulate the combination oscillations. The irreversible forces F” are of the vortical type. The simplest structure of F” and Fd as functionals of X is F”=KX and F”=FX with an antisymmetrica! matrix K and symmetrical matrix F’. In accord with F” and F” one can separate the energy flux W into the vortical W” and dissipative Wd parts. In the case of one-component one-valued X the forces F” vanish and only the dissipative type of the energy transfer realizes. The vortical and dissipative energy fluxes differing in time symmetry have different symmetry with respect to the frequencies Q of the oscillations X, their phases and the values of the combination resonance detunings. The vortical flux is sensitive to the phases of the oscillations X components in a larger degree because interchange of all the phase differences of the oscillations X does not influence W” but replaces W” by WV. Due to this property of the vortical combination forces the multiwave regimes with phases coordinated in a certain manner become most preferable energetically. The increments for such vortical resonances are always larger than for combination nonvortical ones, with ignored W”, which in particular corresponds to the views from the threewave scheme, Different dependences with respect to the resonance detunings mean, with reference to the reso—
2 May 1988
nances (1), that W” is even and Wd is odd with respect to the differences w— co~.So, the contribution to the increments due to the vortical processes is of positive sign (for the favorable phases of the oscillations X) independent of the translation w towards the spectrum {cok}. In conditions of overlappings between the spectral bands {cok} and {co} of the waves able to interact the flux Wd being odd with respect to w— Wk decreases and vanishes while the flux W” can double its intensity. The vortical processes in such conditions are expected to be predominant. Below the simplest display of the vortical phenomena in the stimulated Raman scattering in a transparent medium far from the regions of dispersion is considered and possibilities of considerable increasing of the scattering are pointed to.
2. Display in the stimulated Raman scattering 2.1. Problem statement Following the standard model (see e.g. refs. [2,31) widely used for description of the stimulated Raman scattering the material relation between the polarization P= (P,, P2, P3) and the field E= (Er, E2, E3) is adopted to be linear and local: P(r t)=a,k(x)Ek(r,t)
,
(2)
where a = {a,k} is a real symmetrical tensor of polarizability depending on a set of variables x_{xa(r,t)} characterizing the oscillatory medium state, the index a numerates the oscillations of different nature and polarization. Let us examine the energy flux from the high frequency field to the oscillations x. The expression for its mean density is ~
,,
~
,~
., ~.
9xa and < > denotes time averwhere aika = ôaik/t aging over the scale 1 /Q. The threshold ofthe stimulated Raman scattering corresponds to max $(S_R)civ~O, (4) x
63
Volume 129, number 1
PHYSICS LETTERS A
where the integral is taken over the medium volume, R determines the losses in the subsystem x (for E= 0), it is nonnegative, R ~ 0, and taken even with respect to the replacement x(r, t) by x(r, —t). The variation in (4) is performed on the set of the oscillations x accessible in the system and selects the oscillations with maximum increments which are just realized in the stimulated scattering. Let us consider the combinations resonances (1) keeping in S, eq. (3), the terms quadratic in x. The quadratic approximation is sufficient for determination of the threshold and the increments of the stimulated Raman scattering for small oscillations x. Let the scattering medium occupy the layer —L~z~
exp[i(kkz—wt)]}.
( qz —
cot) 1
(5)
where XqQ=X~Q,_Qand Xq~{XaqQ}. Solving the Maxwell equations for the system with arbitrary functions x(z, t) in the layer polarizability a(x) to the first order in x and substituting the solution into (3) the following expression for the desired functional of the total energy flux is obtained, 64
3 —
=
S(~) dz L
—
2~Q(w+Q) aikaainpXaqQX~.QG,knqq.a ~
6
1
In this sum the indices i, k, n run over the values 1 and 2, the set (a, q, Q) is as in (5). The factors G depend on the wave synchronism detunings ô~= k7 ±(kk + q), where k~= [(w +Q) Ic] n• and have the form Gjkfl4QQ
sin &~qL5~flô,~nq~ L ökq öi~q s~nÔ~qL5~flö~q’ L
+
ôjkq
öjnq
2.3. Vortical and dissipative fluxes The functional (6) includes both fluxes, JJv +r’. The partition can be easily done since the spectraldensity Jv (Q) must be odd and I” (Q) even with respect to Q. The expressions for Ivand ~ contain Q dependent factors D~~qqQ = ~Q[(W+Q)Giknqq’Q ±(w—Q)G,k~,_~._~’._Q] . —
(7)
The factors D~stand in and D in I’~. It is convenient to deal with nonnegative values of Q. Turning to this representation it is not difficult to make sure that the wave energy conversion into the anti-Stokes spectral band is proportional to the first summands in (7) and the conversion into the Stokes band is proportional to the second ones with the minus sign. The expression for the dissipative flux is proportional to the differences between the two factors and the part of J1I connected with the Stokes factors is a nonnegative quadratic in x form, while the part of jd connected withthe anti-Stokes factors is a nonpositive one. That is, the scattered Stokes waves cause a positive contribution to I’’, i.e. provoke the stimulated scattering, and the anti-Stokes waves cause the opposite effect. So, the behaviour of the dissiJV
Here e= (e,, e2, 0), kk= (co/c)nk and n~= 1 + 4ita kk~c is the velocity of light, the 3-axis is chosen along z and the 1- and 2-axes are in line with the two other main directions of a°. The oscillations X {Xa(Z, t) } will be presented as the expansion ~ Xq~exp [i
1
2 May 1988
Volume 129, number I
PHYSICS LETTERS A
pative flux I” corresponds to the guide given by the three-wave model, The expression for Jv is of off-beat character. It is proportional to the factors D i.e. both Stokes and anti-Stokes processes contribute in Jv with the same sign. And the sign of the quadratic in x form Jv depends on the phase relations in {XaqQ} and can be arbitrary. Since w>> Q, ID + I>> ID and the vortical processes can be very important. Let us compare the fluxes JV and jd for the particular case k~L<<1, i= 1, 2. One then obtains, within accuracy —~ (k1L ) 2 ~,
—
1d
JV=
2g~$X~qQX~*qQ sin qL sin q L q q sin qL sin q’L _i(/)QKaflXaq0Xp*q~Q q q’ _Q
,
(8)
where the summation (or integration) over Q is over the range Q~0and g~= —aikaa,np Re(eke~) cn 1 2it = aklT~a1~~ Im (e~ e~) ,
The right-hand sides are the sums over the repeated dummy indices. It is easy to verify that g={gap} is a symmetrical nonnegative definite matrix. It follows that the flux jd in (8) is nonpositive whatever the parameters a~ka, the amplitudes and phases of e and x are. So, in the considered case the combination resonance processes of the dissipative type cannot provoke the stimulated scattering. On the contrary, they cause deexcitation of the combination oscillations. The account of the vortical flux I” can change the situation completely. It is seen from the following. The matrix i,c= {~Kafl} is a hermitian antisymmetrical matrix which is not sign definite. Therefore the quadratic in x form I” in (8) is real valued but is not of definite sign. We can always, if K~0, make Jv> 0 for appropriately chosen phases of the components XaqQ. The dissipation function R in (4) is even with respect to Q, as well as its integral and the flux 1”. Both are even with respect to the phases arg ( XaqQX~q’~ while is an odd function of them. Therefore the maximum increments of the oscillaJV
2 May 1988
tions take place for the phases making JV>o. Note, then, that in general the elements of the matrices g and K are of the same order. In the latter case and if the subsystem x has a degenerate spectrum the maximum increments take place for such oscillations x which minimize 11” I and maximize Jv and for their ratio one arrives at the estimate JV(Q\/Jd
‘
‘
Q
Q
,.~ ‘~
(0
The ratio w/Q can be 102 and more. The more the ratio is the more the vortical processes can prevail over the dissipative ones. the combination the type areSo, expected to be veryresonances importantof for thevortical stimulated Raman scattering in thin layers. The increments caused by the vortical flux can considerably, up to the ratio w/Q, predominate over the decrements caused by a the dissipative flux. The direct analysis of the increments confirms these conclusions. Note that K=0 and consequently 1V0 for onecomponent x, the same holds for linearly polarized incident light. This becomes clear since in the considered phenomena the multi-dimensional high frequency system is the light waves of the two lateral polarizations and the action of the vortical energy transfer mechanism is connected with mixing of the waves of different polarizations. For the effectiveness of the vortical processes the material relation a(x), its linear in x part, must have the anisotropic character connected with rotation of the optical indicatrix in response to variations of x. For the deformations of the indicatrix, without rotation, ,C= 0. Distinctive features of the vortical combination effects are their sharp responses to the degree of the ellipticity of the incident light and the rarefaction of the natural frequencies of the low frequency subsystem. For thick layers conditions of the phase synchronism of the resonantly combined waves become essential restricting the number ofthe interacting wave triples. Nevertheless, possibilities of significant display of the vortical combination resonance processes remain. A detailed analysis will be presented elsewhere. In conclusion, it is worth to compare the (wellknown for the stimulated Raman scattering) partition into the two-photon and parametric processes with the partition into the vortical and dissipative 65
Volume 129, number 1
PHYSICS LETTERS A
processes. The former partition does not show signs of the time symmetry and peculiarities of the multidimensionality. It proceeds from the opposite procedure: from the solutions of the motion x equations as an explicit functional of the field E thus making the interacting energy ~aikaxaEiEka “four-wave” term. In the second order perturbation theory it separates the processes of the stimulated scattering at the Stokes and anti-Stokes frequencies while vortical as well as dissipative fluxes contain terms belonging to both the spectral bands. The vortical combination processes and the way of —
66
2 May 1988
thinking in terms of their separation seem to be important when dealing with effects of coordinated multiwave regimes of the combination resonances.
References [1] V.E. Shapiro, Zh. Eksp. Teor. Fiz. 89 (1985) 1857; 91(1986) 1280 [Soy. Phys. JETP 62 (1985) 1128; 64 (1986) 756]. [2] S.A. Akhmanov and R.V. Khokhlov, Problemi nelineinoi optiki (Moscow, 1964) [in Russian]. [3] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965).
PHYSICS LETTERS A
2 May 1988
COMBINATION RESONANCES OF VORTICAL TYPE V.E. SHAPIRO Kirensky Institute of Physics, Krasnoyarsk 660036, USSR Received 2 November 1987; accepted for publication 26 February 1988 Communicated by V.M. Agranovich
A vortical mechanism of energy transfer between systems with different frequencies of motion is investigated, which performs the energy conversion due to the combination resonances differing from the conventional concept. General remarks and the simplest considerations of the vortical mechanism display in stimulated Raman scattering are presented. The possibility of a considerable increase of the scattering is pointed to.
1. Introduction In stimulated Raman scattering and many other phenomena of high frequency effects in multidimensional systems one meets with concurrence and interaction of a great number of resonantly combined waves. For the combination resonance of a given order the minimum number of waves, corresponding to the number of quanta in the elementary reactions of energy transfer along the spectrum, plays an important role in the interactions. But not all the basic processes inherent in the resonance energy exchange are reproduced with account of only the minimum number of waves. In fact, only one type of process is reproduced which, for considerations explained below, will be called dissipative. There exists a different type of combination resonance process called vortical, which, in a certain sense, is complementary to the dissipative one [1]. Relying on the results of ref. [1] the present note aims at extending this point. Let, for example, only the first order combination resonances (0 = 0~k+
Q,
~
=
c0k —
Q
be a factor, i.e. waves of frequency
(1) co excite
in a me-
dium oscillations of low frequencies Q and waves of Stokes and anti-Stokes frequencies Wk arise. It is often believed for this case that modelling of the wave reservoir by three nonlinearly coupled waves (oscillators) of frequencies co, Wk and Q re62
produces the main resonance processes, that buildup of the combination oscillations arises due to the resonances (1) for which co> ~0k and those forwhich w < COk cause the opposite effect, suppress the oscillations, and that at least in a qualitative aspect this guide is satisfactory for weakly nonlinear systems. But these assertions need revision for systems in which both high and low frequency forms of motion involved in the interactions are multi-dimensional and have a degenerate spectrum. Under such conditions the mentioned vortical combination processes become significant. They tend to involve a number of triples of resonantly combined waves into the coordinated motion and lead to the trend of the resonance phenomena is quitescheme inexplicable from the position of thewhich three-wave of the analysis. In particular, for weakly nonlinear systems, within the three-wave terms in the interaction energy accounted in the second order perturbation theory, build-up of the combination oscillations becomes possible for all values of w including w< min{wk}. The idea of the combination resonance processes with off-beat character may be suggested from symmetry considerations. Let X= (X 1 (t), X2(t), ...) be the variables charactensing the oscillatory (at low frequencies Q) medium state of different spatial forms and F= (F1, F2, ...) be the combination forces conjugated to X. Per cycle of periodic change in X the forces F perform the work
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 129, number I
PHYSICS LETTERSA
W ~‘F~dXi. Here and further summation over repeated indices is implied. The forces F are functions of the scattered high frequency field. Since this field itself is brought about by the oscillations X and a retardation takes place the forces F are a retarded functional of X. In the general case F can be divided into three parts: F= F~ + F +F d where F’ includes all the reversible part of F, which does not contribute to W. The irreversible forces F v are even and F” are odd with respect to the change of the direction ofthe motion Xinto the opposite one, The irreversible forces F’~changing sign when the motion X inverts its direction are forces like friction and dissipation. When representing the negative friction they stimulate the combination oscillations. The irreversible forces F” are of the vortical type. The simplest structure of F” and Fd as functionals of X is F”=KX and F”=FX with an antisymmetrica! matrix K and symmetrical matrix F’. In accord with F” and F” one can separate the energy flux W into the vortical W” and dissipative Wd parts. In the case of one-component one-valued X the forces F” vanish and only the dissipative type of the energy transfer realizes. The vortical and dissipative energy fluxes differing in time symmetry have different symmetry with respect to the frequencies Q of the oscillations X, their phases and the values of the combination resonance detunings. The vortical flux is sensitive to the phases of the oscillations X components in a larger degree because interchange of all the phase differences of the oscillations X does not influence W” but replaces W” by WV. Due to this property of the vortical combination forces the multiwave regimes with phases coordinated in a certain manner become most preferable energetically. The increments for such vortical resonances are always larger than for combination nonvortical ones, with ignored W”, which in particular corresponds to the views from the threewave scheme, Different dependences with respect to the resonance detunings mean, with reference to the reso—
2 May 1988
nances (1), that W” is even and Wd is odd with respect to the differences w— co~.So, the contribution to the increments due to the vortical processes is of positive sign (for the favorable phases of the oscillations X) independent of the translation w towards the spectrum {cok}. In conditions of overlappings between the spectral bands {cok} and {co} of the waves able to interact the flux Wd being odd with respect to w— Wk decreases and vanishes while the flux W” can double its intensity. The vortical processes in such conditions are expected to be predominant. Below the simplest display of the vortical phenomena in the stimulated Raman scattering in a transparent medium far from the regions of dispersion is considered and possibilities of considerable increasing of the scattering are pointed to.
2. Display in the stimulated Raman scattering 2.1. Problem statement Following the standard model (see e.g. refs. [2,31) widely used for description of the stimulated Raman scattering the material relation between the polarization P= (P,, P2, P3) and the field E= (Er, E2, E3) is adopted to be linear and local: P(r t)=a,k(x)Ek(r,t)
,
(2)
where a = {a,k} is a real symmetrical tensor of polarizability depending on a set of variables x_{xa(r,t)} characterizing the oscillatory medium state, the index a numerates the oscillations of different nature and polarization. Let us examine the energy flux from the high frequency field to the oscillations x. The expression for its mean density is ~
,,
~
,~
., ~.
9xa and < > denotes time averwhere aika = ôaik/t aging over the scale 1 /Q. The threshold ofthe stimulated Raman scattering corresponds to max $(S_R)civ~O, (4) x
63
Volume 129, number 1
PHYSICS LETTERS A
where the integral is taken over the medium volume, R determines the losses in the subsystem x (for E= 0), it is nonnegative, R ~ 0, and taken even with respect to the replacement x(r, t) by x(r, —t). The variation in (4) is performed on the set of the oscillations x accessible in the system and selects the oscillations with maximum increments which are just realized in the stimulated scattering. Let us consider the combinations resonances (1) keeping in S, eq. (3), the terms quadratic in x. The quadratic approximation is sufficient for determination of the threshold and the increments of the stimulated Raman scattering for small oscillations x. Let the scattering medium occupy the layer —L~z~
exp[i(kkz—wt)]}.
( qz —
cot) 1
(5)
where XqQ=X~Q,_Qand Xq~{XaqQ}. Solving the Maxwell equations for the system with arbitrary functions x(z, t) in the layer polarizability a(x) to the first order in x and substituting the solution into (3) the following expression for the desired functional of the total energy flux is obtained, 64
3 —
=
S(~) dz L
—
2~Q(w+Q) aikaainpXaqQX~.QG,knqq.a ~
6
1
In this sum the indices i, k, n run over the values 1 and 2, the set (a, q, Q) is as in (5). The factors G depend on the wave synchronism detunings ô~= k7 ±(kk + q), where k~= [(w +Q) Ic] n• and have the form Gjkfl4QQ
sin &~qL5~flô,~nq~ L ökq öi~q s~nÔ~qL5~flö~q’ L
+
ôjkq
öjnq
2.3. Vortical and dissipative fluxes The functional (6) includes both fluxes, JJv +r’. The partition can be easily done since the spectraldensity Jv (Q) must be odd and I” (Q) even with respect to Q. The expressions for Ivand ~ contain Q dependent factors D~~qqQ = ~Q[(W+Q)Giknqq’Q ±(w—Q)G,k~,_~._~’._Q] . —
(7)
The factors D~stand in and D in I’~. It is convenient to deal with nonnegative values of Q. Turning to this representation it is not difficult to make sure that the wave energy conversion into the anti-Stokes spectral band is proportional to the first summands in (7) and the conversion into the Stokes band is proportional to the second ones with the minus sign. The expression for the dissipative flux is proportional to the differences between the two factors and the part of J1I connected with the Stokes factors is a nonnegative quadratic in x form, while the part of jd connected withthe anti-Stokes factors is a nonpositive one. That is, the scattered Stokes waves cause a positive contribution to I’’, i.e. provoke the stimulated scattering, and the anti-Stokes waves cause the opposite effect. So, the behaviour of the dissiJV
Here e= (e,, e2, 0), kk= (co/c)nk and n~= 1 + 4ita kk~c is the velocity of light, the 3-axis is chosen along z and the 1- and 2-axes are in line with the two other main directions of a°. The oscillations X {Xa(Z, t) } will be presented as the expansion ~ Xq~exp [i
1
2 May 1988
Volume 129, number I
PHYSICS LETTERS A
pative flux I” corresponds to the guide given by the three-wave model, The expression for Jv is of off-beat character. It is proportional to the factors D i.e. both Stokes and anti-Stokes processes contribute in Jv with the same sign. And the sign of the quadratic in x form Jv depends on the phase relations in {XaqQ} and can be arbitrary. Since w>> Q, ID + I>> ID and the vortical processes can be very important. Let us compare the fluxes JV and jd for the particular case k~L<<1, i= 1, 2. One then obtains, within accuracy —~ (k1L ) 2 ~,
—
1d
JV=
2g~$X~qQX~*qQ sin qL sin q L q q sin qL sin q’L _i(/)QKaflXaq0Xp*q~Q q q’ _Q
,
(8)
where the summation (or integration) over Q is over the range Q~0and g~= —aikaa,np Re(eke~) cn 1 2it = aklT~a1~~ Im (e~ e~) ,
The right-hand sides are the sums over the repeated dummy indices. It is easy to verify that g={gap} is a symmetrical nonnegative definite matrix. It follows that the flux jd in (8) is nonpositive whatever the parameters a~ka, the amplitudes and phases of e and x are. So, in the considered case the combination resonance processes of the dissipative type cannot provoke the stimulated scattering. On the contrary, they cause deexcitation of the combination oscillations. The account of the vortical flux I” can change the situation completely. It is seen from the following. The matrix i,c= {~Kafl} is a hermitian antisymmetrical matrix which is not sign definite. Therefore the quadratic in x form I” in (8) is real valued but is not of definite sign. We can always, if K~0, make Jv> 0 for appropriately chosen phases of the components XaqQ. The dissipation function R in (4) is even with respect to Q, as well as its integral and the flux 1”. Both are even with respect to the phases arg ( XaqQX~q’~ while is an odd function of them. Therefore the maximum increments of the oscillaJV
2 May 1988
tions take place for the phases making JV>o. Note, then, that in general the elements of the matrices g and K are of the same order. In the latter case and if the subsystem x has a degenerate spectrum the maximum increments take place for such oscillations x which minimize 11” I and maximize Jv and for their ratio one arrives at the estimate JV(Q\/Jd
‘
‘
Q
Q
,.~ ‘~
(0
The ratio w/Q can be 102 and more. The more the ratio is the more the vortical processes can prevail over the dissipative ones. the combination the type areSo, expected to be veryresonances importantof for thevortical stimulated Raman scattering in thin layers. The increments caused by the vortical flux can considerably, up to the ratio w/Q, predominate over the decrements caused by a the dissipative flux. The direct analysis of the increments confirms these conclusions. Note that K=0 and consequently 1V0 for onecomponent x, the same holds for linearly polarized incident light. This becomes clear since in the considered phenomena the multi-dimensional high frequency system is the light waves of the two lateral polarizations and the action of the vortical energy transfer mechanism is connected with mixing of the waves of different polarizations. For the effectiveness of the vortical processes the material relation a(x), its linear in x part, must have the anisotropic character connected with rotation of the optical indicatrix in response to variations of x. For the deformations of the indicatrix, without rotation, ,C= 0. Distinctive features of the vortical combination effects are their sharp responses to the degree of the ellipticity of the incident light and the rarefaction of the natural frequencies of the low frequency subsystem. For thick layers conditions of the phase synchronism of the resonantly combined waves become essential restricting the number ofthe interacting wave triples. Nevertheless, possibilities of significant display of the vortical combination resonance processes remain. A detailed analysis will be presented elsewhere. In conclusion, it is worth to compare the (wellknown for the stimulated Raman scattering) partition into the two-photon and parametric processes with the partition into the vortical and dissipative 65
Volume 129, number 1
PHYSICS LETTERS A
processes. The former partition does not show signs of the time symmetry and peculiarities of the multidimensionality. It proceeds from the opposite procedure: from the solutions of the motion x equations as an explicit functional of the field E thus making the interacting energy ~aikaxaEiEka “four-wave” term. In the second order perturbation theory it separates the processes of the stimulated scattering at the Stokes and anti-Stokes frequencies while vortical as well as dissipative fluxes contain terms belonging to both the spectral bands. The vortical combination processes and the way of —
66
2 May 1988
thinking in terms of their separation seem to be important when dealing with effects of coordinated multiwave regimes of the combination resonances.
References [1] V.E. Shapiro, Zh. Eksp. Teor. Fiz. 89 (1985) 1857; 91(1986) 1280 [Soy. Phys. JETP 62 (1985) 1128; 64 (1986) 756]. [2] S.A. Akhmanov and R.V. Khokhlov, Problemi nelineinoi optiki (Moscow, 1964) [in Russian]. [3] N. Bloembergen, Nonlinear optics (Benjamin, New York, 1965).