Self-induced transparency for love waves

Volume 120, number 2

PHYSICS LETTERS A

2 February 1987

SELF-INDUCED TRANSPARENCY FOR LOVE WAVES G.T. ADAMASHVILI Tbilisi State University, Tbilisi, USSR Received 11 October 1986; accepted for publication 31 October 1986

A theory of self-induced transparency is constructed for the Love wave in a system which consists of a film lying on a semispace in the presence of paramagnetic impurities. The solution is obtained in the form of the 2n-pulse of McCall and H a h n for the Love wave. It is shown that the characteristic parameters of the nonlinear Love wave depend on the transverse-mode profile.

1. The Love wave is a basic type of surface wave with horiontal polarization. These waves are propagating in a semispace on which a thin layer with dissimilar elastic properties is spread. At present the high-frequency Love wave is realized and broadly used in physical experiments. If the system in which such waves are propagating contains paramagnetic impurities, Love waves will interact with them. If the conditions of ref. [ 1 ] are fulfilled self-induced transparency for these waves may occur. The case of paramagnetic impurities evenly distributed in one of the connected media is thoroughly investigated in ref. [21. The situation is changed considerably when paramagnetic impurities are contained in the transition layer on the boundary dividing the film and the semispace. The question whether self-induced transparency exists for the Love wave has not been answered in the latter case up to now. In this letter the question is answered positively and it is shown that in this case we get a qualitatively new result: the 2n-pulse of McCall and Hahn (soliton) for the Love wave. Note, that in ref. [ 2 ] the results were obtained in the form of a pulsating soliton (breathers) for these waves. 2. Let us consider the effect of self-induced transparency for the Love wave in a system which consists of a plane-parallel layer lying on the elastic semispace. As an example of a simple model we take the case of two non-metallic, diamagnetic crystals with cubic 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

symmetry and dissimilar elastic properties. In the boundary dividing these media we assume that there are paramagneticc impurities in small concentration, with effective spin S = ½. Suppose that an external constant magnetic field Hot tz is applied along one of the axes of the fourth order (the x and y axes are in both crystals directed along the other axes of the fourth order). Let us select the boundary of the plane between the layer and the elastic semispace as the (x, y) plane; furthermore we assume that to the semispace corresponds the region z < 0 (index 1 ), and to the layer corresponds the region 0 < z < h (index 2). Consider the case of a wave with wave vector k(0, Q, x), propagating along the y axis, and of which the vector of deformation u = (u, 0, 0) is directed along the x axis. For simplicity, we suppose that none of the quantities depend on the x coordinate. Later on the quantities attributed to both media will be written without index. The equation of the acoustic field is

O2u Oa:,y + drr'~z P o t z - 3~---fiZz' crJ'z=axz+#xz'

(1)

where p is the density of the crystal, axy and trxz are the stress tensors, components of the connected media, and &:,~ is the contribution to this quantity, caused by the presence of paramagnetic impurities. We shall seek the components of the deformation vector in the following form:

73

Volume 120, number 2

PHYSICS LETTERS A

2 February 1987

which can be o b t a i n e d from the h a m i l t o n i a n (wilh //=.~.+ × e "~' .... ~ do) d k ,

(2i

where the f u n c t i o n f ( w , k: z) defines the transversem o d e profile. In the b o u n d a r y - v a l u e p r o b l e m s o f the theory of elastic waves the b o u n d a r y conditions play a dominating role. I f both m e d i a are rigidly interconnected over the interface, in any point o f this surface the d e f o r m a t i o n vector u should obviously be continuous during the transition from one m e d i u m to the other. This c o n d i t i o n can be written as [ 3 ] ul ° > = u ~ °' ,

i3)

(where the index " 0 " implies that the value has been taken at z = O ) . Along with this, on the d i v i d i n g surface the condition o f stress continuity should be satisfied (as z = 0), which in the presence of a transient layer can be represented by o-~o) . 7.

= o.~o2

(4)

In the last equation, the fact that p a r a m a g n e t i c impurities are contained in the transient layer is taken into account; this layer occupies the space 0 >t z > - h,~ (analogously the case where the transient layer occupies the space O<~z<~ho can be c o n s i d e r e d ) . It has been assumed that the thickness o f the transition layer ho << 2, where it is the wavelength. In this approxim a t i o n all the quantities in the space 0 >i z > - h o can be considered to be i n d e p e n d e n t o f z and therefore we can ignore the thickness o f the transient layer. On the free b o u n d a r y o f the layer, the following b o u n d a r y condition is satisfied: a2_,.~ 1= h = 0 .

51

The quantity &t.,= is defined from the Bloch equations (in the a p p r o x i m a t i o n o f rotating waves, considering the time o f irreversible relaxation T> Tl-~): OS ~ Ot _½i~a(S

~+-S+~

74

(~}

where .Zl-=oJoS: is the h a m i l t o n i a n o f the Zeeman interaction..J¢, p = f l F t t o S ' ~ ,. is the h a m i l t o n i a n of the interaction of the spin system with the Love wave, ¢0o is the Zeeman frequency o f the spins, fl is the Bohr magneton. F = F ..... - - F .... are the components o f the s p i n - p h o n o n coupling tensors, ~ ~,: = e * + e are the components o f the deformation tensor [ 1 ], : f = / ] H , k and S ' = ½(5;' +.5" ). The solution o f eq. (6) is the case of exact resonance ,Q= ¢o<~and is of the form u=0.

c-}N.

sinO.

-

½NocosO.

18

where /

O(y,t)=Uj

e(y,t'ldt'

(9

is the pulse area of the Love wave at z = 0 and N,, is the concentration o f active particles at z = 0. Quantities interpreted in (8) and (9) are determined from the following relations: = ( u + i t : ) e

''~" ">"'

e'(y,t)=e(y.t)e""e'

,(S > = N ,

c~,>

!10)

where
"

?(')"

a ~ = ½'~"< ,7' > = _+ ~ i ~ ? v ; ,

s i n O e ' "*~'

w,

( 1I )

3. Substituting expression (2) into (1) and taking into account eqs. ( 3 ) and ( 5 ) we obtain the function which defines the transverse-mode profile: f(oJ, k; z ) = e x p [ K l ( ( O , k ) z ] ,

2<0 ,

=tg[#%(o), k)h]sin[Ke(oJ, k ) z ]

),

OS + Ot - i~ooS + - i50e + S : ,

~g, p.

+cos[x2(oJ, k)z], (6)

0
Volume 120, n u m b e r 2

PHYSICS LETTERS A

with

2 February 1987

_( OF(o), k)"I

x,(o), k) = ( k 2 -- o)2]C2) 1/2 ,

/£2(09, k) = ( o)2 /c22 - k 2) ,/2 ,

(F~o)o-\ (13)

defining the velocity of the damping of the surface wave along the z axis, where c,, 2 are the transverse polarized sound velocities of the connected media. From the boundary condition (4) follows the equation which is applicable when z = O:

Oo) p2 h

f o - - Pl/£1

£2fo

]o~=a.k=o--

c~ '

/£22 "~-J~2I C 2l /C 22

cosZ/£2h +

/£12/£22

'

(20)

_(OF(o),k)'] (F'k)o - \ Ok 1,o =a, k=O

hp2c

f f F(o), k)~lxz(o),k)e i(ky-'°t) do)dk

=

-or)

\ ~2K-~2

+ plC~/£1 co"--$2/£2h, J '

(21)

F(o), k) = 1 p , c ~ x , ( o ) , k ) t g [ / £ 2 ( o ) , k ) h l ,

(15)

with v the velocity of the linear wave. Eq. (18 ) is the well-known sine-Gordon equation which is investigated in ref. [ 5 ] by means of the inverse scattering method. For simplicity we shall revert to the variable z = t-y/vo from eq. (18 ); then it is easy to obtain the well-known equation (vo is the velocity of pulse)

{lxz(O), k ) = 1/£ 1((J), k)~l(O), k ) ,

(16)

d20 - : T -2 sin O dz2

= - - [ 7 - - / 4 p l c2 ,

(14)

where

p2C2/£2(o), k)

(22)

° )l

The series expansion of the function F(o), k) with respect to the frequency £2 and the "wave vector" Q of the carrier waves reads [ 2,4 ]

- 16pl£2fo

- 1

,

(23)

where T is the pulse width. The solution of eq. (22) is the well-known 2n-pulse of McCall and Hahn (soliton) for the Love wave at

F(o), k)=F(£2, Q)

+ OF(o), k)/Oo) Io~=a, k=Q(o) --£2)

z=0:

+ OF(O), k)/Okl~o=a, k = o ( k - Q) + .... e (z) = (2/~WT) sech (z/T) . Substituting this expansion into eq. (14) and taking into account relations (10) and (1 1 ), we obtain after separating real and imaginary parts the dispersion law for the Love wave Pl c21/£~(£2' Q) tg[/£2(£2, Q)hl-p2c~/£2(£2, Q)

(17)

and the nonlinear equation describing the change in the envelope of the acoustic wave e (y, t) (at z = 0) 020

--

Ot 2

+v

020

Ot Oy

=-a

2 sin O ,

(18)

where v=

(FDo (F/o) o '

a2=

~2No - 16p, £2fo '

(19)

Expressions (2), (10), (12) and (13) define the transverse-mode profile of the soliton. From (19), (20) and (21) it is clear that (F~,)o > 0, (FL)o < 0 and therefore v> 0, the width of the soliton T is real and positive. Hence, in the situation being investigated at the execution of the inequalities £2T>> 1, T< c2 < v < c,, vo< v, the experimental observation of the 27t-pulse of McCall and Hahn for the Love wave can be carried out. From relations (20) and (23) follows that the soliton delay time in a medium, at the data width, is dependent upon th~ transverse-mode profile. This is a qualitatively new result in comparison with the result obtained in an unbounded medium [ 1 ]. In conclusion, let us note that untill now, only the case of self-induced transparency for the volume 75

Volume 120, number 2

PHYSICS LETTERS A

acoustic w a v e s has b e e n e x p e r i m e n t a l l y investigated, w h e r e the m a t e r i a l s used are for e x a m p l e the crystals M g O a n d L i N b O 3 a c t i v a t e d by the Fe ~ + ions ( f o r M g O also N i 2~ ) with e f f e c t i v e spins S = 1 [ 1,6 ]. Besides, along with the case o f r e s o n a n c e o n b , ~ = 0 ~ o , there has also b e e n s t u d i e d the p r o p a g a t i o n o f w a v e s by acoustic self-induced transparency u n d e r c o n d i t i o n s o f inexact r e s o n a n c e , a n d in the p r e s e n c e o f an i n h o m o g e n e o u s b r o a d e n i n g o f the lines A M P . G e n e r a l i z a t i o n o f the a b o v e t h e o r y for the c o n d i tions which h a v e b e e n realized in the e x p e r i m e n t s o f refs. [ 1,6] d o e s not c o n t a i n any sort o f specific hardships. T h e r e f o r e we are not going to generalize, as this does not lead to any q u a l i t a t i v e l y n e w result in c o m p a r i s o n w i t h the w a v e a l r e a d y specified above.

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2 Februa~ 1987

References [1 ] N.S. Shiren. Phys. Re,,. B 2 (It>70) 2471. 12l G.T. Adamashvili, Tr. Tbilis. Gos. [Jniv. Fiz. 244 (1983) 5 Solid State Commun. 47 ( 1982 t 497. L.D. Landau and E.M. I_ifshilz, l'heorya uprugost) ( Nauka Moscow, 1965 ). [ 4 ] V.M. Agranovich. V.I. Rupasov and V. Ya. Chernyak, Pls'ma Zh. Eksp. Teor. Fiz. 33 (1981) 196. [5]V.E. Zakharov, S,V. Manakov, S.P. Novikox and I.P Pitaevskii, Theoria solitonov ( Nauka, Moscow, 1980 ). [6] V.V. Samartsev, B.P. Smoliakov and R.Z. Sharipov, Pis'ma Zh. Eksp. Toot. Fiz. 20 (19741 644