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Intrinsic optical bistability for low-frequency rigid layer modes in MoS2 crystal
a _-
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22 January 19% PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 210 (1996) 416-422
Intrinsic optical bistability for low-frequency rigid layer modes in MoS, crystal Boiena Ratajska-Gadomska, Wojciech Gadomski Laboratory of Physicochemistry
of Dielectrics and Magnetics, Department of Chemistry, Warsaw Uniuersity, ul. i!wirki i Wigury 101, 02-089 Warsaw, Poland
Received 8 August 1995; revised manuscript received 25 October 1995; accepted for publication 17 November 1995 Communicated by A. Lagendijk
Abstract
A study of the parametric optical excitation of overtones of rigid layer modes in a MO& crystal is presented. This effect, resulting from the modulation of the vibrational frequency by a biharmonic optical field, is shown to lead to a significant intrinsic optical bistability, which can be registered in a four-wave mixing process,
The effect of optical b&ability still attracts the interest of scientists due to its applicability in optoelectronical devices. From this point of view elements based on intrinsic (mirrorless) bistability seem to be easier in practical use, since they do not need any additional external cavity. Theoretical models predicting the occurrence of intrinsic bistability have taken into account increasing absorption, interatomic correlation in a many-body quantum model [I], anharmonicity in a classical driven Duffing oscillator [2] and dipole-dipole interaction between two level atoms driven by an external field [3]. The first one was demonstrated experimentally [S]. In our previous papers [5] we have presented the theoretical classical model of the concrete physical system, i.e., anharmonic crystal lattice modes parametrically driven by a biharmonic optical field E(r) = E, cos( w,t) + E, cos( w,t), in which intrinsic bistability is due to the interaction between dipoles optically induced in molecules (atoms, ions) composing the lattice. As a result, the vibrational frequency of the mode becomes modulated with the difference frequency w1 - w2, whereas the modulation coefficient 6 is proportional to the external field amplitudes E, and E2 and inversely proportional to the frequency of the crystal mode oO, The frequencies of the lattice modes considered lie in the infrared region, o, - lo’*- 1013 s- ‘. 5 = KE, E,/w,. For frequencies of the incident fields close to the resonance condition o, - o2 = 2 w,, a coherent overtone vibration is generated. Its stationary amplitude exhibits a strong bistability with respect to the input field intensities and the detuning of the difference frequency from the resonant value. Since it is proportional to 5 [5], the strongest enhancement of the overtone amplitude occurs for low-frequency modes. It has been shown that this effect can be detected in a four wave-mixing process [5] and in the nonlinear refractive index of a crystal [6]. In order to estimate the order of magnitude of the parameters, which might be observed experimentally, numerical calculations have been performed for a few crystal lattices of different structural forces, i.e. a homopolar lattice of diamond, a molecular crystal of benzene [5], and recently for a rigid layer mode in layered 03759601/96/$12.00
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B. Ratajska-Gadomska. W. Gadomski/ Physics Letters A 210 (19%) 416-422
TO - mode
LO - mode
G axis
sg
,$yxis<2g;
4
;I;
E
5
;I
I ,
(a) Fig. 1. Configuration
of the ions in one layer of a 2H-MO&
Fig. 2. (a) The primitive cell and (b) the corresponding
(a)
(b)
(b)
crystal. The filled circles denote MO ions, and the open circles denote S ions.
linear chain model of a
2H-MoS, crystal, for the LO mcde and the TO mode,
crystal GaSe [7]. It appeared that the modulation coefficient is two orders of magnitude larger for the low-frequency LO rigid layer mode in a GaSe crystal than for crystals of higher symmetry like, e.g., diamond. This result suggests that, due to their special symmetry connected with the weak interlayer forces, layered crystals might be good nonlinear materials for bistable devices. In view of this suggestion we continue the search of such materials among layered crystals. In this paper we present the calculations performed for all (LO and TO) rigid layer vibrations of a 2H-MoS, crystals, for different polarizations of optical fields towards the direction of vibration. Molybdenum sulphide is one of the most intensely investigated layer compounds as it exists in nature and can be easily obtained. Fig. 1 shows the coordination of ions in a single layer. The frequencies of the incident optical fields are chosen to lie in the transparency region of the crystal, which spreads from 0.68 pm to 13 pm 181. The 2H polytype of MoS, has the space group D& and its primitive cell contains six atoms belonging to two identical layers (Fig. 2a). The ions of the same type create the planes perpendicular to a crystal axis (c-axis). We consider two long-wavelength rigid layer modes, i.e., the LO mode, in which the planes vibrate rigidly toward each other parallel to the c-axis, and the TO mode, in which the planes slide over each other perpendicular to the c-axis (see Fig. 2). Those vibrations can be well described by a linear-chain model [9], in which ions, representing the whole plane, lie along the chain being the c-axis of the crystal (see Fig. 2b). Since the intralayer forces of ionic and homopolar character are about 100 times stronger than interlayer forces of van der Waals nature [lo], the intralayer forces between MO and S ions are described as harmonic oscillators, whereas the forces between S atoms belonging to adjacent layers are described as anharmonic oscillators. Taking into account the symmetry relations in the crystal and limiting mutual interactions between the ions to the nearest neighbours, we obtain the structural Hamiltonian of the crystal in the form
&I = $a[(u2
-
u,)2+ (243- u2)2+ (Uj
+;A,[(u,-u~)‘+(u~--~)~]
-
u3)2+ (ug - LJ,>2] + &[(
+tA4[(~,-~(6)~f(~3-~4)~],
u, -
u$ + (ug -
14J2] (1)
where ui denotes the displacement of the ith ion (see Fig. 2) from the equilibrium position, C, and C, are the corresponding harmonic force constants, A, and A, are the anharmonic coefficients for interlayer interaction. The frequencies of the incident optical fields are taken in the range of transparency of the crystal. We apply the model of interaction introduced previously [5], in which the dipoles induced by an optical field in all ions
B. RatajsbGaabmska,
418
W. Gadomski/Physics
Letters A 210 (19%) 416-422
interact not only with an external field but also with the field of dipoles induced in neighbouring ions. Thus, the effective local field acting on each ion (In), where 1 is the number of an elementary cell and n is the number of a molecule in a cell, is given by E;“( In) =fii&
cos( wlr) +fZiEZi cos( tizt) + c Aij( In, l’n’)E;“( /In’),
(24
jl'n'
where fki is a Lorentz local-field factor, fki = $[ ni( w,j + 21 (n,.( wj is the refractive index of a crystal) and Ti( In, l’?QL( In, f’n) Aij( In, l’n) = c akj( l’n’)F3( In, 1%‘)
-
r2( En, M)
k
P4
4,
is the (iln, J’n’) element of the matrix of dipole-dipole interactions, whereas we consider only the nearest neighbours (l’n’j of ion (in). okI denotes the polarizability of the ion (In> and r,(ln, I’d) = r&n, I’d> + ui(h) - u,(l’n’) is the vector connecting atoms (In> and (Pn’j. The effective field acting on each ion (In) is obtained as the solution of Eq. (2aj, E:“(h)
= C(Z--A),~‘(ln,
l’d)[f,,~,,
cos(o,t)
+f,i~,i
cos(w,t)f,;],
(3)
jl'n'
where I denotes the unity matrix and (I - A); ’ (In, l’n’) is the (iln, jl’ n’) element of the inverse of the matrix I-A. The interaction Hamiltonian of the system is considered in the form H, = - c a ( In) EL”( ln) ET”( In) .
(4
lnij
Substituting Eq. (3) into Eq. (4) and taking advantage of translational symmetry of a crystal we get H,=-;z
2 ijkmn-I
i
i n’= 1
1=0.-1.1
X{Jw,,.f,,f,,
aij(n)(Z-A),‘(On,
ln’)(Z-A)j;,‘(On,
I’d’)
n” = 1 r-o.-I.1
+&%mf*kfZm
+ cLfZm~1k~2m
+f,m.&k‘%&k)
co++
-
%DlI* (5)
Here,weareconsidering
two q = 0 optical lattice vibrations B& and E$. For both the ionic displacements fulfill the following conditions, -uq=
u,=u3=
-uug=u,
u*=
2GmMo -uu5=u,
u=” i I+ Ca(mMo+2ms)
i
’
0,‘=
4Cb mMo +
%
where mMo and ms are the masses of molybdenum and sulphur ions, respectively. In the B$ mode ions vibrate along the chain with u 11c (Fig. 2a) and frequency CIJ~ = 56 cm-‘, whereas in the Eis mode ions vibrate in the direction perpendicular to the chain with u _Lc (Fig. 2b) and frequency Ok = 33.7 cm-‘. The dynamics of the crystal mode, which resonantly interacts with the optical field, is described in terms of dimensionless amplitudes x, y, z defined as x=(IQ1*)/(1Q,I’~,
y=/
z=(ldl*)/(lQ,~“~~
where Q and Q are the normal mode amplitude and momentum, respectively, ( > denotes averaging over all other modes of the crystal and ( 1Q, I* > IS . the mean square amplitude of the considered mode in the absence of external fields. In the case considered the normal amplitude Q has the form 1 ‘=2
[
‘+’
(
2CPMO ‘+ Ca(mMo+2ms)
iI’
B. Ratajska-Gadomska,
W. Gadomski/
419
Physics Letters A 210 11996) 416-422
16000c 14000 =
*
1
04
1
,
i
i
;:’ ,
’
/, ,:.
,.;
12000 -
1
J
g 10000 Nw" 8000 -
:'
i
3
6000 ;’ 4000 -
2.10
2.05
2.20
2.15
2.25
,I
2.05
2.10
v
i
2.15 v
2.20
2.25
Fig. 3. (a) The dimensionlessamplitude a of the overtone vibration and(b) the correspondingintensity of tbe output field I E3(2w, - q)\ * versus relative frequency v = (0, - 02)/w, for 15,1)E2 (1c; (1) LO mode, E, = 2COOesu and E, = 100 esu; (2) TO mode, E, = 2000 esu and E2 = 100 esu; (3) TO mode, E, = Zoo0 esu and E, = 400 esu; (4) LO mode in a diamond lattice [5] for the same fields as in (1).
The time evolution of the system is described by the following equations [51, k=2z,
jJ= -2z[l-
.$ COS(VT-a)]
i=y-x[l-~cos(VT-8)]
-~EXZ-4y(y-
l),
-EXZ-22yy
(6)
1Q, I 2> and y are the mode nonlinearity and damping coefficient respectively, is the modulation frequency of the mode, &= KE, E2/wa is the modulation coefficient y= (01 - O&W* dependent on external fields, representing the ratio of the force due to dipole-dipole interactions to the structure force in the crystal. The solution of Eq. (6) is obtained in the form [5]
where
E = (16A,/m,)(
X(T) =a(7)
cos[vr+p(7)]
+b(r).
(7) The amplitude a(r) is the dimensionless amplitude of the overtone of frequency v = 2 + ~0, where ~0 is the small detuning dependent on the crystal nonlinearity, and b(t) is the dimensionless dc-amplitude. The stationary solutions of both amplitudes u, and &, exhibit strong bistability with respect to the detuning from the overtone frequency and with respect to the intensities of the input fields (Figs. 3a, 4a, 5a, 6a). Since the mean polarizability gij of the whole crystal can be expressed as a power series in the vibrational amplitudes [ 111, ~ij=~~~‘+~sq’:‘(a)(~.)+~C~aj,)(qa,-qa)(Ia,,Iz~. CX 4a
(8)
the resonant enhancement of the overtone amplitude a, results in the nonlinear crystal polarization at frequency
2 OJ,- ti2, given by the formula [5] ‘i(t)
=&
C~~~‘(a)(IQ,,l’>a,(r)fiElj J
COS[(~W, -~2)t+~(t)],
where V is the volume of the crystal and (Ydenotes the chosen from other modes, being out of resonance, has been neglected.
q =
(9)
0 resonant mode, whereas the contribution
420
B. Rarajska-Gadomskn,
0
W. Gadomski/Physics
2000
Leners A 210 (1996) 416-422
6000 4000 6ooo I E,E,I (‘IO3 esu)
loo00
Fig. 4. Tbe dimensionless amplitude a of the overtone vibration and the corresponding intensity of the output field 1E,(2 w , - 02) I* versus input field intensities for E, 11E2 11c; (1) LO mode and v = 2.15, (2) TO mode and Y = 2.213 (Y lying in the bistability region of the curves in Fig. 3).
The optical field E3 at frequency 2~0, - w2. generated in a crystal, reflects the bistable behaviour of the vibrational amplitude (Figs. 3b, 4b, 5b, 6b). Here we have applied the model of a crystal polarizability Pij presented in Ref. [12], in which it is assumed to be the sum of the effective polarizabilities of the ions composing the lattice,
where the effective polarizability ai”if’(ln) is the polarizability of an isolated ion, optically induced in the nearest neighbours, ai”j”(Zn)=
c
aij(h>,
modified by dipoles
(11)
an’),
aik(Zn)(l-A)i;(Zn,
kl’ n’
N ;” -
6000
-
4000
-
0.05 0.00 7 16
2.18
2.20 V
2.22
2.24
2.16
2.18
2.20
2.22
2.24
V
Fig. 5. The dimensionless amplitude a of the overtone vibration and the corresponding intensity of tbe output field 1,?I,(2 0, - w,) I * versus relative frequency Y =(o, - o~)/u+, for the TO mode for E, II E2 I c; (1) E, = 2000 esu and ES = 100 esu; (2) E, = 2000 esu and E, = 200 esu; (3) E, = 2000 esu and E, = 400 esu.
B. Ratajska-Gadomska,
W. Gad&&i/Physics
421
Letters A 210 (1996) 416-422
amplitude a of the overtone vibration and tbe corresponding intensity of the output field ( E,(20, - 02) I* versus the input field intensities for the TO mode for E, 11E, I c and v = 2.216 (v lying in the bistability region of the curves in Fig. 5).
Fig. 6. The dimensionless
The values of cuij(ln> have been estimated from the Lorentz-Lorentz formula. In this paper we present the calculations performed for two polarizations of applied optical fields, i.e. parallel to the c-axis, E, I(E2 IIc, and perpendicular to the c-axis, E, IIE, _Lc. The values of the calculated parameters are presented in Tables 1 and 2, respectively. For comparison, in the third column of Table 1 we have presented the parameters calculated in Ref. [7] for the LO mode in a GaSe crystal. The modulation parameter for the LO modes in GaSe and MoS, crystals are comparable and one order of magnitude larger than for the TO mode. Fig. 3 shows the dispersive bistability of the vibrational amplitudes and the corresponding output field intensities ) E&2 wi - w2) I * for two rigid layer modes, for both input fields parallel to the c-axis. It can be seen that, for the same values of the input field intensities, in the case of the LO mode B& kxu-ve (1) in Figs. 3a, 3b.)
Table 1 Calculated
modulation
coefficients
coeffkient
for MoS, for E, IIE2 II c compared
t/E,
E2
crystal nonlinearity E P’2’(0, a)( I Qo(a) I ’ > Lkar refractive index n damping coefficient y polarizabilities of isolated ions:
akO 4’
Table 2 Calculated
coeffkients
for MoS,
modulation
coefficient
t/E,
with those for GaSe [7]
LO mode B&
TO mode E&
GaSe, LO mode A”2
3.5 X 10m8 esu 0.089
3.7 X 10e9 esu 0.16 2.5 X low2 esu
3X10-8esu 0.14
7 X lo-* esu 2.5 0.01 2.38 x lO-24 cm3 3.89 x lO-24 cm3
2.5 0.01 2.38 X lO-24 cm3 3.89 X lO-24 cm3
6X lo-* 2.65 0.01
esu
for E, II E, I c LO mode B&
E2
X lo- ‘* esu 0.089 2.3 X lo-’ esu 3.88 0.01 3.09 X lO-24 cm3 5.14 X lO-24 cm3
-7.15
crystal nonlinearity E P;;)(O. a)( I Q,,(a) I *> linear refractive index rt damping coefficient y polarizabilities of isolated ions: CYII;, al’
TO mode E$
-4.76~ lO-9 esu 0.16 1.5 X 10m3 esu 3.88 0.01 3.09 x 10-24 cm3 5.14 X lO-24 cm3
422
B. Ratajska-Gadomska,
W. Gadomski/Physics
Letters A 210 11996) 416-422
the overtone amplitude and the output field are a few times stronger than in the case of the TO mode E& (curve (2) in Fig. 3a). C urves (3) in Figs. 3a,3b correspond to the stronger external field applied for the mode E&. For comparison we have also shown the overtone amplitude, which would be generated in a diamond lattice h-ve (4) in Fig. 3a) by the external fields of the same values as in curves (1) and (2). The b&able behaviour of the overtone amplitude and the output field with respect to the input field intensities for both modes considered are shown in Fig. 4. In the case of both external fields perpendicular to the c-axis (Figs. 5 and 6) the effect is negligible for the LO mode (see the numerical value of 5 in Table 1). Thus, Fig. 5 shows the dispersive bistability of the overtone amplitude and of the generated field intensity I E,(2w, - w2) I ’ only for the TO mode, for three different values of one of the input fields. The bistability of the overtone amplitude and the output field with respect to the external field amplitudes are shown in Fig. 6, for the case corresponding to curve (3) in Fig. 5. The TO vibration is shifted in phase by 7r/2 with respect to the LO vibration, i.e., formula (4) has the form x(r) = a(r) cos[ ~7 + P(T) + 7r/2] + b(r). Comparison of Figs. 4 and 5 proves that the configuration of the LO mode and the external fields parallel to the c-axis (perpendicular to the layer planes) is most convenient for observation of intrinsic optical bistability in layered crystals. Moreover, it is worthwhile to underline that the B& mode in a MoS, crystal is infrared-inactive and Raman-inactive so that overtone spectroscopy provides a good tool for the estimation of this mode. This work was supported by KNB grant P302 134 07.
References [l] C.M. Bowden and F.H. Hopf, in: Optical instabilities, eds. R.W. Boyd, M.G. Raymer and L.M. Narducci (Cambridge Univ. Press, Cambridge, 1986). and references therein. [2] C. Flytzanis and C.S. Tang, Phys. Rev. Len. 45 (1980) 441. [3] F.H. Hopf and CM. Bowden. Phys. Rev. A 32 (1985) 268. [4] H. Rossman, F. Henneberger and J. Voight, Phys. Stat. Sol. B 115 (1983) K63. [5] B. Ratajska-Gadomska and W. Gadomski, in: Optical instabilities, eds. R.W. Boyd, M.G. Raymer and LM. Narducci (Cambridge Univ. Press, Cambridge, 1986); W. Gadomski and B. Ratajska-Gadomska, Phys. Rev. A 34 (1986) 1277. [6] B. Ratajska-Gadomska and W. Gadomski, Phys. Lett. A 117 (1986) 156. [7] B. Ratajska-Gadomska and W. Gadomski, Appt. Opt. 34 (1995) 4326. [S] J.A. Wilson and A.D. Yoffe, Philos. Mag. 25 (1972) 625; W.W. Sobolev and W.Wal. Sobolev, Solid State Phys. 36 (1994) 2560. [9] T.J. Wieting, Solid State Commun. 12 (1973) 931. [lo] T.J. Wieting and J.L. Verble, in: Electrons and phonons in layered crystal structures, eds. T.J. Wieting and M. Schluter (Reidel, Dordrecht, 1979) p. 383. [ 1 l] M. Born and K. Huang, Dynamical theory of crystal lattice (Clarendon, [12] B. Ratajska-Gadomska, Phys. Rev. B 26 (1982) 1942.
Oxford,
1954).
_l!fz
22 January 19% PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 210 (1996) 416-422
Intrinsic optical bistability for low-frequency rigid layer modes in MoS, crystal Boiena Ratajska-Gadomska, Wojciech Gadomski Laboratory of Physicochemistry
of Dielectrics and Magnetics, Department of Chemistry, Warsaw Uniuersity, ul. i!wirki i Wigury 101, 02-089 Warsaw, Poland
Received 8 August 1995; revised manuscript received 25 October 1995; accepted for publication 17 November 1995 Communicated by A. Lagendijk
Abstract
A study of the parametric optical excitation of overtones of rigid layer modes in a MO& crystal is presented. This effect, resulting from the modulation of the vibrational frequency by a biharmonic optical field, is shown to lead to a significant intrinsic optical bistability, which can be registered in a four-wave mixing process,
The effect of optical b&ability still attracts the interest of scientists due to its applicability in optoelectronical devices. From this point of view elements based on intrinsic (mirrorless) bistability seem to be easier in practical use, since they do not need any additional external cavity. Theoretical models predicting the occurrence of intrinsic bistability have taken into account increasing absorption, interatomic correlation in a many-body quantum model [I], anharmonicity in a classical driven Duffing oscillator [2] and dipole-dipole interaction between two level atoms driven by an external field [3]. The first one was demonstrated experimentally [S]. In our previous papers [5] we have presented the theoretical classical model of the concrete physical system, i.e., anharmonic crystal lattice modes parametrically driven by a biharmonic optical field E(r) = E, cos( w,t) + E, cos( w,t), in which intrinsic bistability is due to the interaction between dipoles optically induced in molecules (atoms, ions) composing the lattice. As a result, the vibrational frequency of the mode becomes modulated with the difference frequency w1 - w2, whereas the modulation coefficient 6 is proportional to the external field amplitudes E, and E2 and inversely proportional to the frequency of the crystal mode oO, The frequencies of the lattice modes considered lie in the infrared region, o, - lo’*- 1013 s- ‘. 5 = KE, E,/w,. For frequencies of the incident fields close to the resonance condition o, - o2 = 2 w,, a coherent overtone vibration is generated. Its stationary amplitude exhibits a strong bistability with respect to the input field intensities and the detuning of the difference frequency from the resonant value. Since it is proportional to 5 [5], the strongest enhancement of the overtone amplitude occurs for low-frequency modes. It has been shown that this effect can be detected in a four wave-mixing process [5] and in the nonlinear refractive index of a crystal [6]. In order to estimate the order of magnitude of the parameters, which might be observed experimentally, numerical calculations have been performed for a few crystal lattices of different structural forces, i.e. a homopolar lattice of diamond, a molecular crystal of benzene [5], and recently for a rigid layer mode in layered 03759601/96/$12.00
Q 19% Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00913-2
417
B. Ratajska-Gadomska. W. Gadomski/ Physics Letters A 210 (19%) 416-422
TO - mode
LO - mode
G axis
sg
,$yxis<2g;
4
;I;
E
5
;I
I ,
(a) Fig. 1. Configuration
of the ions in one layer of a 2H-MO&
Fig. 2. (a) The primitive cell and (b) the corresponding
(a)
(b)
(b)
crystal. The filled circles denote MO ions, and the open circles denote S ions.
linear chain model of a
2H-MoS, crystal, for the LO mcde and the TO mode,
crystal GaSe [7]. It appeared that the modulation coefficient is two orders of magnitude larger for the low-frequency LO rigid layer mode in a GaSe crystal than for crystals of higher symmetry like, e.g., diamond. This result suggests that, due to their special symmetry connected with the weak interlayer forces, layered crystals might be good nonlinear materials for bistable devices. In view of this suggestion we continue the search of such materials among layered crystals. In this paper we present the calculations performed for all (LO and TO) rigid layer vibrations of a 2H-MoS, crystals, for different polarizations of optical fields towards the direction of vibration. Molybdenum sulphide is one of the most intensely investigated layer compounds as it exists in nature and can be easily obtained. Fig. 1 shows the coordination of ions in a single layer. The frequencies of the incident optical fields are chosen to lie in the transparency region of the crystal, which spreads from 0.68 pm to 13 pm 181. The 2H polytype of MoS, has the space group D& and its primitive cell contains six atoms belonging to two identical layers (Fig. 2a). The ions of the same type create the planes perpendicular to a crystal axis (c-axis). We consider two long-wavelength rigid layer modes, i.e., the LO mode, in which the planes vibrate rigidly toward each other parallel to the c-axis, and the TO mode, in which the planes slide over each other perpendicular to the c-axis (see Fig. 2). Those vibrations can be well described by a linear-chain model [9], in which ions, representing the whole plane, lie along the chain being the c-axis of the crystal (see Fig. 2b). Since the intralayer forces of ionic and homopolar character are about 100 times stronger than interlayer forces of van der Waals nature [lo], the intralayer forces between MO and S ions are described as harmonic oscillators, whereas the forces between S atoms belonging to adjacent layers are described as anharmonic oscillators. Taking into account the symmetry relations in the crystal and limiting mutual interactions between the ions to the nearest neighbours, we obtain the structural Hamiltonian of the crystal in the form
&I = $a[(u2
-
u,)2+ (243- u2)2+ (Uj
+;A,[(u,-u~)‘+(u~--~)~]
-
u3)2+ (ug - LJ,>2] + &[(
+tA4[(~,-~(6)~f(~3-~4)~],
u, -
u$ + (ug -
14J2] (1)
where ui denotes the displacement of the ith ion (see Fig. 2) from the equilibrium position, C, and C, are the corresponding harmonic force constants, A, and A, are the anharmonic coefficients for interlayer interaction. The frequencies of the incident optical fields are taken in the range of transparency of the crystal. We apply the model of interaction introduced previously [5], in which the dipoles induced by an optical field in all ions
B. RatajsbGaabmska,
418
W. Gadomski/Physics
Letters A 210 (19%) 416-422
interact not only with an external field but also with the field of dipoles induced in neighbouring ions. Thus, the effective local field acting on each ion (In), where 1 is the number of an elementary cell and n is the number of a molecule in a cell, is given by E;“( In) =fii&
cos( wlr) +fZiEZi cos( tizt) + c Aij( In, l’n’)E;“( /In’),
(24
jl'n'
where fki is a Lorentz local-field factor, fki = $[ ni( w,j + 21 (n,.( wj is the refractive index of a crystal) and Ti( In, l’?QL( In, f’n) Aij( In, l’n) = c akj( l’n’)F3( In, 1%‘)
-
r2( En, M)
k
P4
4,
is the (iln, J’n’) element of the matrix of dipole-dipole interactions, whereas we consider only the nearest neighbours (l’n’j of ion (in). okI denotes the polarizability of the ion (In> and r,(ln, I’d) = r&n, I’d> + ui(h) - u,(l’n’) is the vector connecting atoms (In> and (Pn’j. The effective field acting on each ion (In) is obtained as the solution of Eq. (2aj, E:“(h)
= C(Z--A),~‘(ln,
l’d)[f,,~,,
cos(o,t)
+f,i~,i
cos(w,t)f,;],
(3)
jl'n'
where I denotes the unity matrix and (I - A); ’ (In, l’n’) is the (iln, jl’ n’) element of the inverse of the matrix I-A. The interaction Hamiltonian of the system is considered in the form H, = - c a ( In) EL”( ln) ET”( In) .
(4
lnij
Substituting Eq. (3) into Eq. (4) and taking advantage of translational symmetry of a crystal we get H,=-;z
2 ijkmn-I
i
i n’= 1
1=0.-1.1
X{Jw,,.f,,f,,
aij(n)(Z-A),‘(On,
ln’)(Z-A)j;,‘(On,
I’d’)
n” = 1 r-o.-I.1
+&%mf*kfZm
+ cLfZm~1k~2m
+f,m.&k‘%&k)
co++
-
%DlI* (5)
Here,weareconsidering
two q = 0 optical lattice vibrations B& and E$. For both the ionic displacements fulfill the following conditions, -uq=
u,=u3=
-uug=u,
u*=
2GmMo -uu5=u,
u=” i I+ Ca(mMo+2ms)
i
’
0,‘=
4Cb mMo +
%
where mMo and ms are the masses of molybdenum and sulphur ions, respectively. In the B$ mode ions vibrate along the chain with u 11c (Fig. 2a) and frequency CIJ~ = 56 cm-‘, whereas in the Eis mode ions vibrate in the direction perpendicular to the chain with u _Lc (Fig. 2b) and frequency Ok = 33.7 cm-‘. The dynamics of the crystal mode, which resonantly interacts with the optical field, is described in terms of dimensionless amplitudes x, y, z defined as x=(IQ1*)/(1Q,I’~,
y=
z=(ldl*)/(lQ,~“~~
where Q and Q are the normal mode amplitude and momentum, respectively, ( > denotes averaging over all other modes of the crystal and ( 1Q, I* > IS . the mean square amplitude of the considered mode in the absence of external fields. In the case considered the normal amplitude Q has the form 1 ‘=2
[
‘+’
(
2CPMO ‘+ Ca(mMo+2ms)
iI’
B. Ratajska-Gadomska,
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419
Physics Letters A 210 11996) 416-422
16000c 14000 =
*
1
04
1
,
i
i
;:’ ,
’
/, ,:.
,.;
12000 -
1
J
g 10000 Nw" 8000 -
:'
i
3
6000 ;’ 4000 -
2.10
2.05
2.20
2.15
2.25
,I
2.05
2.10
v
i
2.15 v
2.20
2.25
Fig. 3. (a) The dimensionlessamplitude a of the overtone vibration and(b) the correspondingintensity of tbe output field I E3(2w, - q)\ * versus relative frequency v = (0, - 02)/w, for 15,1)E2 (1c; (1) LO mode, E, = 2COOesu and E, = 100 esu; (2) TO mode, E, = 2000 esu and E2 = 100 esu; (3) TO mode, E, = Zoo0 esu and E, = 400 esu; (4) LO mode in a diamond lattice [5] for the same fields as in (1).
The time evolution of the system is described by the following equations [51, k=2z,
jJ= -2z[l-
.$ COS(VT-a)]
i=y-x[l-~cos(VT-8)]
-~EXZ-4y(y-
l),
-EXZ-22yy
(6)
1Q, I 2> and y are the mode nonlinearity and damping coefficient respectively, is the modulation frequency of the mode, &= KE, E2/wa is the modulation coefficient y= (01 - O&W* dependent on external fields, representing the ratio of the force due to dipole-dipole interactions to the structure force in the crystal. The solution of Eq. (6) is obtained in the form [5]
where
E = (16A,/m,)(
X(T) =a(7)
cos[vr+p(7)]
+b(r).
(7) The amplitude a(r) is the dimensionless amplitude of the overtone of frequency v = 2 + ~0, where ~0 is the small detuning dependent on the crystal nonlinearity, and b(t) is the dimensionless dc-amplitude. The stationary solutions of both amplitudes u, and &, exhibit strong bistability with respect to the detuning from the overtone frequency and with respect to the intensities of the input fields (Figs. 3a, 4a, 5a, 6a). Since the mean polarizability gij of the whole crystal can be expressed as a power series in the vibrational amplitudes [ 111, ~ij=~~~‘+~sq’:‘(a)(~.)+~C~aj,)(qa,-qa)(Ia,,Iz~. CX 4a
(8)
the resonant enhancement of the overtone amplitude a, results in the nonlinear crystal polarization at frequency
2 OJ,- ti2, given by the formula [5] ‘i(t)
=&
C~~~‘(a)(IQ,,l’>a,(r)fiElj J
COS[(~W, -~2)t+~(t)],
where V is the volume of the crystal and (Ydenotes the chosen from other modes, being out of resonance, has been neglected.
q =
(9)
0 resonant mode, whereas the contribution
420
B. Rarajska-Gadomskn,
0
W. Gadomski/Physics
2000
Leners A 210 (1996) 416-422
6000 4000 6ooo I E,E,I (‘IO3 esu)
loo00
Fig. 4. Tbe dimensionless amplitude a of the overtone vibration and the corresponding intensity of the output field 1E,(2 w , - 02) I* versus input field intensities for E, 11E2 11c; (1) LO mode and v = 2.15, (2) TO mode and Y = 2.213 (Y lying in the bistability region of the curves in Fig. 3).
The optical field E3 at frequency 2~0, - w2. generated in a crystal, reflects the bistable behaviour of the vibrational amplitude (Figs. 3b, 4b, 5b, 6b). Here we have applied the model of a crystal polarizability Pij presented in Ref. [12], in which it is assumed to be the sum of the effective polarizabilities of the ions composing the lattice,
where the effective polarizability ai”if’(ln) is the polarizability of an isolated ion, optically induced in the nearest neighbours, ai”j”(Zn)=
c
aij(h>,
modified by dipoles
(11)
an’),
aik(Zn)(l-A)i;(Zn,
kl’ n’
N ;” -
6000
-
4000
-
0.05 0.00 7 16
2.18
2.20 V
2.22
2.24
2.16
2.18
2.20
2.22
2.24
V
Fig. 5. The dimensionless amplitude a of the overtone vibration and the corresponding intensity of tbe output field 1,?I,(2 0, - w,) I * versus relative frequency Y =(o, - o~)/u+, for the TO mode for E, II E2 I c; (1) E, = 2000 esu and ES = 100 esu; (2) E, = 2000 esu and E, = 200 esu; (3) E, = 2000 esu and E, = 400 esu.
B. Ratajska-Gadomska,
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421
Letters A 210 (1996) 416-422
amplitude a of the overtone vibration and tbe corresponding intensity of the output field ( E,(20, - 02) I* versus the input field intensities for the TO mode for E, 11E, I c and v = 2.216 (v lying in the bistability region of the curves in Fig. 5).
Fig. 6. The dimensionless
The values of cuij(ln> have been estimated from the Lorentz-Lorentz formula. In this paper we present the calculations performed for two polarizations of applied optical fields, i.e. parallel to the c-axis, E, I(E2 IIc, and perpendicular to the c-axis, E, IIE, _Lc. The values of the calculated parameters are presented in Tables 1 and 2, respectively. For comparison, in the third column of Table 1 we have presented the parameters calculated in Ref. [7] for the LO mode in a GaSe crystal. The modulation parameter for the LO modes in GaSe and MoS, crystals are comparable and one order of magnitude larger than for the TO mode. Fig. 3 shows the dispersive bistability of the vibrational amplitudes and the corresponding output field intensities ) E&2 wi - w2) I * for two rigid layer modes, for both input fields parallel to the c-axis. It can be seen that, for the same values of the input field intensities, in the case of the LO mode B& kxu-ve (1) in Figs. 3a, 3b.)
Table 1 Calculated
modulation
coefficients
coeffkient
for MoS, for E, IIE2 II c compared
t/E,
E2
crystal nonlinearity E P’2’(0, a)( I Qo(a) I ’ > Lkar refractive index n damping coefficient y polarizabilities of isolated ions:
akO 4’
Table 2 Calculated
coeffkients
for MoS,
modulation
coefficient
t/E,
with those for GaSe [7]
LO mode B&
TO mode E&
GaSe, LO mode A”2
3.5 X 10m8 esu 0.089
3.7 X 10e9 esu 0.16 2.5 X low2 esu
3X10-8esu 0.14
7 X lo-* esu 2.5 0.01 2.38 x lO-24 cm3 3.89 x lO-24 cm3
2.5 0.01 2.38 X lO-24 cm3 3.89 X lO-24 cm3
6X lo-* 2.65 0.01
esu
for E, II E, I c LO mode B&
E2
X lo- ‘* esu 0.089 2.3 X lo-’ esu 3.88 0.01 3.09 X lO-24 cm3 5.14 X lO-24 cm3
-7.15
crystal nonlinearity E P;;)(O. a)( I Q,,(a) I *> linear refractive index rt damping coefficient y polarizabilities of isolated ions: CYII;, al’
TO mode E$
-4.76~ lO-9 esu 0.16 1.5 X 10m3 esu 3.88 0.01 3.09 x 10-24 cm3 5.14 X lO-24 cm3
422
B. Ratajska-Gadomska,
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Letters A 210 11996) 416-422
the overtone amplitude and the output field are a few times stronger than in the case of the TO mode E& (curve (2) in Fig. 3a). C urves (3) in Figs. 3a,3b correspond to the stronger external field applied for the mode E&. For comparison we have also shown the overtone amplitude, which would be generated in a diamond lattice h-ve (4) in Fig. 3a) by the external fields of the same values as in curves (1) and (2). The b&able behaviour of the overtone amplitude and the output field with respect to the input field intensities for both modes considered are shown in Fig. 4. In the case of both external fields perpendicular to the c-axis (Figs. 5 and 6) the effect is negligible for the LO mode (see the numerical value of 5 in Table 1). Thus, Fig. 5 shows the dispersive bistability of the overtone amplitude and of the generated field intensity I E,(2w, - w2) I ’ only for the TO mode, for three different values of one of the input fields. The bistability of the overtone amplitude and the output field with respect to the external field amplitudes are shown in Fig. 6, for the case corresponding to curve (3) in Fig. 5. The TO vibration is shifted in phase by 7r/2 with respect to the LO vibration, i.e., formula (4) has the form x(r) = a(r) cos[ ~7 + P(T) + 7r/2] + b(r). Comparison of Figs. 4 and 5 proves that the configuration of the LO mode and the external fields parallel to the c-axis (perpendicular to the layer planes) is most convenient for observation of intrinsic optical bistability in layered crystals. Moreover, it is worthwhile to underline that the B& mode in a MoS, crystal is infrared-inactive and Raman-inactive so that overtone spectroscopy provides a good tool for the estimation of this mode. This work was supported by KNB grant P302 134 07.
References [l] C.M. Bowden and F.H. Hopf, in: Optical instabilities, eds. R.W. Boyd, M.G. Raymer and L.M. Narducci (Cambridge Univ. Press, Cambridge, 1986). and references therein. [2] C. Flytzanis and C.S. Tang, Phys. Rev. Len. 45 (1980) 441. [3] F.H. Hopf and CM. Bowden. Phys. Rev. A 32 (1985) 268. [4] H. Rossman, F. Henneberger and J. Voight, Phys. Stat. Sol. B 115 (1983) K63. [5] B. Ratajska-Gadomska and W. Gadomski, in: Optical instabilities, eds. R.W. Boyd, M.G. Raymer and LM. Narducci (Cambridge Univ. Press, Cambridge, 1986); W. Gadomski and B. Ratajska-Gadomska, Phys. Rev. A 34 (1986) 1277. [6] B. Ratajska-Gadomska and W. Gadomski, Phys. Lett. A 117 (1986) 156. [7] B. Ratajska-Gadomska and W. Gadomski, Appt. Opt. 34 (1995) 4326. [S] J.A. Wilson and A.D. Yoffe, Philos. Mag. 25 (1972) 625; W.W. Sobolev and W.Wal. Sobolev, Solid State Phys. 36 (1994) 2560. [9] T.J. Wieting, Solid State Commun. 12 (1973) 931. [lo] T.J. Wieting and J.L. Verble, in: Electrons and phonons in layered crystal structures, eds. T.J. Wieting and M. Schluter (Reidel, Dordrecht, 1979) p. 383. [ 1 l] M. Born and K. Huang, Dynamical theory of crystal lattice (Clarendon, [12] B. Ratajska-Gadomska, Phys. Rev. B 26 (1982) 1942.
Oxford,
1954).