CHIM5-SE JOURNAL OF PIIYSICS VOL. 30, NO. 1 FEBRUARY 1992
Analysis of Beam Size Effect on the Laser Induced Frgedericksz
Transition and the Dynamic Response in Nematie Liquid Crystal
(SCB) Films with a Free Surface
Shyu-Mou Chen
Institute of Electro-Opticul Engineering Nutionol Chiuo Tung University, Hsinchu, Tuiwun 30049, Republic of Chinu
und
Ru-Pin Pan
Department of Electrophysics, Nutionul Chiuo Tung University, Hsinchu,
films
Tuiwun 30049, Republic of Chinu (Received December 12,199l)
The non-local effect caused by the finite laser beam size in the laser induced Frdedericksz transition for the homeotropically aligned nematic liquid crystal film is derived analytically and calculated numerically. The influences on the transition threshold and the dynamic response are discussed. The coupling between backflow effect and the non-local effect isshown to be important for the films with a free surface by the calculated results and by comparing with the existing experimental results.
I. INTRODUCTION
Recently, the turn off times of molecular reorientation of nematic liquid crystal (NLC) have been studied in several works.13 The films with a free surface (FS) and the films sandwiched between two glasses, i.e., the so-called hard boundaries (HB) films, have been shown having different turn off times in both of the laser induced Freedericksz transition work’ and the magnetic field induced Freedericksz work.’ The backflow effect and the different flow boundary conditions are considered as the cause of the difference. A careful derivation has been given in Ref. 2 for a uniform magnetic field. However in a more detail measurement on the turn off time caused by varying the intensity of incidental Ar+ laser light,3 it is found that the difference can not be explained by the backflow effect only. Although several studies of the laser-induced molecular reorientation in nematic liquid crystal films have shown that the non-local effect is significant.“*s A pure non-local effect correction, where the angle variation on transverses direction caused by the finite laser beam size is considered, can not explain the dif-ference satisfactorily, either. In this work, we start with the original equation of motion of molecular orientation under a laser field with a Gaussian distribution, then the contribution on
129 cl992 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
-130 ANALYSIS OF BEAh4 SIZE EFFECT ON THE LASER INDUCED... VOL. 30
non-local effect, backflow effect,6 which is caused by the coupling of orientational motion and translational motion, and particularly the coupling of these two effects are derived and dis-cussed.
II. EQUATION OF MOTION
For comparing with experimentsP we consider homeotropical NLC films with thickness ranging from 150 to 300 pm and a linearly polarized Ar + laser beam being normally incidental upon the fdms. The laser beam intensity at the NLC sample is assumed having a symmetrical Gaussian profile, i.e., I@) = I,exp(-p2/&), with 2w equal to 500 pm. We set z-axis along the light propagation direction, and the polarization direction of the laser beam along the x-axis, the director & is (nx, 0, n,) = (sine, 0, cosf3). Here the orientation angle of NLC director, 8, is a function of x, y and z to reflect the finite-size effect on the molecular reorientation.
Following the Ericksen and Leslie’s continuum theory of NLC7 and using the one constant approximation, the coupled differential equations of motion for small angle 6 are obtained as:
CIIe+K(g+v~)e-+
a
Q-V, = 0,az
a. a2
1 3-6 + q,-v, + zff4v:vZ = 0, 8Z dZ2(14
(lb)
where K is the elastic constant, C’ = n,’ [l - (n0’/lr,‘)2]lc, c is the speed of light, n,’ and n,’ are respectively the ordinary and the extraordinary refractive index of NLC with respect to the pump beam wavelength, y1 is the viscosity coefficient for molecular rotation, T]~ = 1/2(cq +a~-a& y1 s (a3-a2) and CZ~, a~..., a6 are the various viscosity coefficients coupling the rotational and translational motion, and V12 is the Laplacian operator in the transverse plane. The term 1/2a4V,2vX in Eq. (lb) hs ows the coupling between the backflow effect and the non-local effect. The detailed derivations are given in the appendix. For neatness of equations, the partial deriva-tives with respect to spatial coordinates will be denoted by & or with “,a” in subscript. For ex-ample, a,2 E a2/az2, and vx;Z = a/& v,.
III. THRESHOLD INTENSITY, Ith
Here, we derive the threshold laser intensity, I,,,, of the laser induced Freedericksz tran-sition for a film with thickness, d. Only when 1, is greater than Ilh, the molecular angle 0(z) is not zero.
In the steady state, 8 = v, = 0, Eq. (1) becomes:
where the cylindrical coordinates @,#J,z) are used.
VOL. 30 SHYU-MOU CHEN AND RU-PIN PAN 131
Due to the radial symmetry of the laser intensity and the one-constant approximation, 0 is independent of #J, i.e., 0 = e(z,p). Using Vt2 = a2J+? + l/p a/+ % VP2, Eq. (2) becomes:
c-‘Z(p)B(p,
2) +zqa;
With the variable separation
@Z(Z) = -q?z(z),
+ qyqp,
2) =
0.technique and setting @,z) = R(p)Z(z), we get:
q2
1
R(P) = 0,(3)
(4a)
(4b)
where q is a constant. For films with a strong homeotropical anchoring112 on the free surface, the boundary conditions for both of the FS samples and the HB samples are 0(z = +d/2) = 0, thus from Eq. (4a), we obtain
Z(z) = cos(qt), (5)
with q = n/d for the lowest order solution, which corresponds to a small angle variation and thus a small energy increasing.
In general, Eq. (4b) can not be solved analytically for a Gaussian type of Z(D). However, L. Csillag et al.’ have introduced an approximation for the Gaussian beam profile and derived the threshold analytically. We summarize their results at follows. In order to obtain the analytic solution, they replaced the Gaussian profile by an effective profile?
(6)
which keeps the power P = Zo(nw2) unchanged. In this situation, the non-trivial solution of Eq. (4b) with the boundaT conditions R(p = 0) = finite, R’@ = 0) =0 and R@ = m) = 0 is:’
R(P) = &J&9,&PZi’0(qP), P > w>P < w,
where/, and K, are the zeroth-order Bessel-function and modified Hankel-function, respective-ly, ~9 is a constant, and A2 = (Zo/CK)-(n/d)2. Solutions of physical meaning are obtained only with A I 0, which implies the existence of a certain threshold. The threshold intensity is then expressed in an empirical form,
Z,, = Z,h,(l + b”k”(“-‘I); (7)
where Zrb = CK(n/d)‘, k = izwld, and the fitted parameters are b = 1.43 and m = 0.24.’ Eq. (4b) with the original Gaussian laser beam profile can be solved easily by numerical method’ for the same boundary conditions without the approximation of Eq. (6) for given w and
132 ANALYSIS OF BEAh4 SIZE EFFECT ON THE LASER INDUCED... VOL. 30
d. The numerical solution of the maximum molecular reorientation angle 8, of Eq. (4b) as a function of the laser intensity I,, for w = 250 pm and d = 200 pm is shown in Fig. (la) as an example. The existence of threshold is obvious from the curve. In Fig. (lb) we show some of
1.20 1 FIG. 1. 1.08 0.96 i 0.84 G 0.72 g 0.60 a9 0.48 0.36 0.24 0.12
-(4
0 45 90 135 180 225 270 315 360 405 450 600 0.0 0.5 1 .O 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 (rr/d)2 (1 04cm2) - - - - noNLE(a) The numerical solution of the maximum molecular reorientation angle 8, as a function of the laser intensity&, for a Gaussian profile intensity with I(’ = 250/cn1 and d = 2OO/tnr; (b) the threshold intensity (Ilh) versus inverse of square of film thickness (1/d2) for )I’ = 100, 200, and 3OO,~tn1, respec-tively. Solution without non-local effect is shown with dashed line.
I
VOL. 30 SHYU-MOU CHEN AND RU-PIN PAN 133
these Z!h as a function of (n/d)‘, for w = 100,200, and 300 pm, respectively. Here we have used no’ = 1.54, n,’ = 1.73,” and K = 0.72 x low6 dyne.” Fitting these points of numerical results with same form as in Eq. (7), we found that b and nt are 1.7 and 0.4, respectively. Thus the threshold intensity estimated with approximation in Eq. (6) is smaller than that with a Gaussian profile by a factor about 1.26, for the cases with w z 250 Pam and,d ranging from 150 to 300 pm. However, Eq. (7) is still a good expression for Irh only that the values for b and m are readjusted as mentioned.
Now, Eq. (7) can be rewritten as Id =
m-*(x/q,
where K* = E;Il +b2k2@-l)], b = 1.7, and nz = 0.4. This the same form for the threshold for a infinite large beam, Itho, only that an effective elastic constant K* is used instead of the true elastic constant K. Since K* is greater than K, the non-local effect makes the threshold intensity increased.
IV. DYNAMIC RESPONSE TIME CONSTANTS, ton AND ~Ofi
If the laser intensity IO is changed abruptly from an initial intensity II smaller than Zth to an intensity IO larger than Zth, the molecules will begin to rotate, we define the beginning ex-ponential time constant as the turn-on time constant, T,. Similarly, if the laser intensity is changed from Z2 larger than It* to Z, smaller than Zth, the molecular orientation will relax to 8 = ‘0, the turn-off time constant, toff, is defined as the exponential time constant at the end. In both cases, the time constant are evaluated at small angles, and B oc e’p. However 7, is positive while tog is negative. In this chapter, we derive these time constants as a function of laser in-tensity I,.
Again, using variable separation:
Vz(t,P,
2) =
v1(t,z)&(p),
(94
(9b)
and substituting into Eq. (l), one obtains:
C-‘I(p)&Re(p) +
Iq3,“(&)Re(P) +
wpe(P)l(lob)
Here we still use the approximation as in Eq. (6) but with the modified values of b and 171 in I!h, then R&p), Vp’R,g@), R,(o) and Vp’R,,@) are same functions ofp except with different constant
134 ANALYSIS OF BEAM SIZE EFFECT ON THE LASER INDUCED... VOL. 30
factors. Here we also assume that I@@) and R,@) have the same form of R@) which is the steady state solution when I, is just above It,,. Eq. (4b) becomes:
V$(p) + c%(p) = 0, (11)
where a2 = (bl~)~(nw/d)~. Eq. (10) is then simplified as:
c-‘1,er + K[af - a*]& - Yrf$ - o2?J1,, = 0, (12a)
1
d,[c& + n&1,*] - --crqa%r = 0.
2 (12b)
For HB case, the following trial solution is used to satisfy the boundary conditions e(z= &d/2) = v& = M/2) = o,r2
el(t, 2) = 0, cos(qz)e’/‘,
vr(t, 2) = v,[sin(qz) - 2t/qe’lT. Substituting into Eq. (12), one obtains:
(134
(13b)
[C-II0 - K(q2 + (22) - Y’]e
l- 0 cos(q.2) - cr2~qcos(qz) - $z& = 0,
$48,
sin(qz) - ncq2v,sin(qr) - iu,02v,[sin(q.z) - $1 = 0.(14a)
(14b)
The variable z is eliminated by integrating from .z = -d/2 to .z = +d/2, after multiplying Eqs. (14a) and (14b) with cos(qz) and sin(qz), respectively, these equations then become
[WIT - fqq* + u”) - F]e, - Q2(q - -$jvo = 0 ,
Wa)
Combining the last two equations, the time constant ZL,HB(I~) is obtained as:
where
(15b)
064
YTL,HB
z
Xi1 -
~(~2Y1s,)[l+
~(ar/qc)(u/q)‘]-‘].
(16b)
The subscript L denotes the finite “Laser” beam size effect to distinguish from a uniform field.For FS sample, the boundary conditions are:12
VOL. 30 SHYU-MOU CHEN AND RU-PIN PAN
o=v,= 0, at z = -d/2; t3 = v,~, = 0, at t = +d/2.
To satisfy these boundary conditions, we choose the trial solutions for Eq. (12) as 01 (t , 2) = 8, cos(qz)e’jT,
vl(t, 2) = uo[l + sin(*z)]ef/T.
By similar work, the response time constant of FS samples, 7,ys(lo), is obtained as
‘G&(L) =
c7*1
(L - h),
lL,FS where 7:L,FS E 7& - (4/71%)[1 + ;(a4/%)(e/4)2]-1}.(19b)
135 (17) Wa) (1Sb) (19a)If the nonlocal effect is neglected, then a = 0, and Y*~L,,HB reduces to Y*I,HB = yl[l -1/6(c~~/y1~,)] and y*lk,~s reduces to y*~,~s = ~~(1 - c~~/ylq~), which are the effective viscosity coefficients in the uniform field. Thus the non-local effect make the effective viscosity coeffi-cient increased.
According to our definitions, both ton and T~Q are with small molecular orientation. Therefore Eqs. (16a) and (19a) can be used for both cases.
If the applied laser intensity is reduced to zero from II, the zero intensity turn off rate
7~~'(0) from Eqs. (16a and 19a) is, r;,:(O) = -(I~*/7;L,i)(~2/d7),
where y*lbi = Y*&HB (or y*~b,~s) for HB (or FS) samples. = -(K/y*l)(n2/d2), with y*l = Y*I,HB = Y*I,FS for HB and
V. RESULTS
(20)
For the case of uniform field, ~~'(0)
FS samples, respectively.
Here we present our calculated results for various situations. Using the parameters for 5CB as in previous sections, the threshold intensity of the case without non-local effect, Iho, as a function of l/d2 is also shown in Fig. (lb) by a dashed line. The non-local effect on the threshold intensity is obvious from this figure.
For comparing the non-local effect and the backflow effect on the dynamic properties, the time constants are calculated with Eq. (I6a) and (19a) for the following 5 situations, and the effective elastic constants and effective viscosities used are shown in the parenthesis:
(1) neglecting both of the two effects, (K, yI); (2) including only non-local effect, (K’, ~1);
136 ANALYSIS OF BEAM SIZE EFFECT ON THE LASER INDUCED... VOL. 30
(3) including only backflow effect, (K, y*&
(4) including both the two effects, but neglecting the non-local effect on the effective vis-cosity, (K*, y*l);
and
(5) including both the two effects and the coupling of these two effects, which is the non-local effect on the effective viscosity, (K*, yolk).
Beside the parameters mentioned earlier, the other parameters for 5CB used are: y1 = 0.68 P, a2 = -0.723 P, a4 = 0.8 P, qc = 1.53~,~ and d = 2OOpm.
The calculated results for l/t versus intensity I, are shown in Fig. 2(a) for FS case. The long dashed line is for situation (l), where neither non-local effect nor backflow effect is con-sidered. The short dashed line is for situation (2), it showed that the pure non-local effect would shift line 1 to the right for a constant value leaving the slope unchanged. The dotted line is for situation (3), it shows a pure backflow effect makes the slope steeper but leaves the threshold unchanged. The alternate dashed line shows the combination of situation (2) and (3), the slope is same as for a pure backflow effect, but the threshold is shifted as a pure non-local effect. With the solid line, the coupling of non-local effect and the backflow effect is added and the line is less steeper than lines 3 and 4 but still steeper than lines 1 and 2.
For the HB case, the relations of these five situations are similar to FS as shown in Fig. 2(b). The shift of threshold is the same as for FS film. However, the influence of backflow is smaller, the coupling with non-local effect is even less significant. Line 4 is almost overlapping with line 5. The coupling effect can be neglected for HB case, and the equations of motion can be simplified once more by dropping the transverse Laplacian term in Eq. (lb).
V I . C O N C L U S I O N
The response time of the free surface film is less than that for the hard boundaries film. When consider backflow only, the ratio of the response time for HB and FS surface (which is same as to comparing the ratio of zero field turn off times at same thickness) is the same as the ratio under a uniform magnetic field.2 Due to the non-local effect caused by the finite size of laser beam and the elastic interaction of liquid crystals, the effective viscosity caused by backflow is modified. We call this the coupling effect between non-local effect and backflow effect. Due to this coupling, we predict the measured ratio as above mentioned should be more closer to 1 than that from the uniform magnetic field experiments. Our example shows that the ratio chan-ges from 1.8 to values between 1.55 and 1.69 for the thickness between 150 and 300 pm. The ratio is slightly dependent on the thickness due to the thickness dependence of YOWL as in Eqs. (16) and (19). This conclusion can explain the recent experimental study on the dynamics of laser induced Freedericksz transition.3
VOL. 30 FIG. 2.
QI
-0.02 -0.04 - 0 . 0 6 - - (l).none - - - - (2).NLE . . . (3)-B= - - - (4).BFE+NLE - (S).BFE+ONE 6C ,->- / .-_.-- / 0 . 0 4 0.03SHYU-MOU CHEN AND RU-PIN PAN 137
-0.02
(1 )-None (2).NLE (3).BFE
(4,5).BFE+NE(or +NLE)
Calculated dynamic response rate (l/t) as a function of the laser intensity (lo) for 5 situations: (1) neglecting both the non-local effect (NLE) and the backflow effect (BFE); (2) including only NLE, neglecting the BFE, (3) including only BFE, neglecting the NLE; (4) including both the two effects, but neglecting the NLE on the effective viscosity; and (5) including both the two effects and the NLE coupling on the effective viscosity. (a) For FS case, and (b) for HB case.
138 ANALYSIS OF BEAh4 SIZE EFFECT ON THE LASER INDUCED... VOL. 30
ACKNOWLEDGMENTS
This work was supported by the National Science Council of the Republic of China under Grant No. NSC79-020%MOO9-30.
APPENDIX: DERIVATIONS OF THE EQUATIONS OF MOTION FOR MOLECULAR REORIENTATION INDUCED BY A LASER BEAM IN
NEMATIC LIQUID CRYSTAL FILMS
We set z-axis along the light propagation directior., the polarization direction of the laser beam alongx-axis, the director n then can be written as: n” = (n,,OpJ = (sirB,O,cos@, here 8 is a function of x, y and z, to reflect the finite-size effect on the molecular reorientation. With the one constant approximation, the elastic free energy density, F, is given by:7
where K is the distortion elastic constant of NLC. The comma in subscript denotes partial dif-ferentiation w.r.t. spatial coordinates, e.g., nqZ = an,/az.
The external applied body force7
G on the director 4 due to the electric field E of the laser beam is: G = .s,/4rr < (4 * E)E >, where E, is the anisotropy of the dielectric constant; E= = (&‘)2 - (&‘)2, n,’ and n,’ are the ordinary and the extraordinary refractive indices, respectively, and E = (E,, 0, Ez). E, is determined by the continuity of the tangential component of E, E,
= E,, where E, is the electric vector of the incoming light beam; Ez is calculated from the con-dition of 2-E.E = 0, which leads to Ez = -l/2E,.s,sin28/[(n,‘)2 - .s,sin28].‘3 Thus,
&
(G,,o,G,)=
E,~S (rib)* sin 6 l,O,- E, sin ~9 cos B4s (n:)” - E. sin2 0 (n:)” - E. sin2 6
.
1
Using the Ericksen and Leslie’s continuum theory of NLC7 assuming the fluid-flow velocity v = (v+,,vJ, with each component vi, i = x, y, and z, being a function of X, y, and z, and neglecting the inertial effect, we can obtain six coupled partial differential equations as fol-lowing:
G, + ysin 0 + li’{cos0V29 - sin 0[(0,,)2 + (0,,)2 _I- (0,,)2]}
(A.2a) -71 CosfJ - 02 COSBV,,, - a3cos0vl,, - -y2sinBv,,, = 0,
VOL. 30 SHYU-MOU CHEN AND RU-PIN PAN 139
G, + y cos 0 + Ii’{ - sin 0V2fI - cos 0[(0,,)” + (e,,)” + (e,,)“]}
(A.2c) +yl sin 8, - a3 sin Bv,,, - cq sin ev,,, - 72 cos Bv~,~ = 0,
t zz,z + tyz,y + tzz,, = 0, (A.2d)
t zy,z + t,,,, + tty,z = 0, ( A . 2 4
t zz,z + +?,y +t t tz,* = 0, (A.29
where
tIZ = --p - K(BJ2 +
1
272 sin(20)e + a(2 al sin2 B - (~2 - c~3 + erg + ag) sin(28)v,,,+:(2nl sin2 8 + a2 + a3 + a5 + a6) sin(20)v,,,
+[a~ sin4 0 + a4 + (a5 + a6) sin2 0]v,,, + $I sin2(20)v,,,
t yo = -K(~,z)(~,y) + :[a4 + (a6 + 03) Sin2 @i,y 1
+-[a4 + ( a 6 - a3) Sin2 e]vy,z 2
+;(a6 - a3) Sin(28)vy,z + :(a6 + a3) Sin(28)vz,yl t zz = -K(B,,)(B,,) + (02 cos2 0 - a3 sin2 0)e
1 1
$-(-a1 Sin2 28 - a 2 COS2 ti + a3 Sin2 0 + a4 + a5 cOS2 0 + a6 Sin2 e)vz,z 2 2
1 1
+ ;( ;crl sin2 2~9 + a2 cos2 0 - a3 sin2 6 + a4 + a5 c0S 2 0 + a6 Sin2 e)vz,z
‘ ‘
+:(a~ sin2 0 + 05) sin(2+,,, + ;(a1 cos2 0 t zy = -h’(B,,)(B,,) + :[a4 + (a5 + a2) Sin2
4vc,y
+:[a, + (a5 -
c~2) sin2 B]vy,z+ a6) Sin(2+z,z,
+:(a5 - a2) sin(2+y,, + :(a5 + a2) Sin(2%,y, t Y Y = -p - qe,y)2 + Q47Jy,y,
140 ANALYSIS OF BEAM SIZE EFFECT ON THE LASER INDUCED... VOL. 30
t,, = -Iqe,,)(e,,) +
gLl +
(a5 + a2)cos2
B]VZJ+;[a4 +
(a5 - 02)co2
Q&z+:(a5 +
a2) sin(2e)v,,y + :(a5 - a2) sin(2f+y,z, t,, = -h’(e,,)(e,,) + (-cr2 sin’ e + a3 ~0s~ e)i+:(:a1 sin2 28 - a2 sin2 e + a3 60s~ e + a4 + a5 sin2 e + a6 ~0s~ e)v,,,
+;(a1 sin2 e + ~6) sin(2B)v,,, + ~(cK~ ~0s~ e + CX~) sin(2f+,,, ,
t,, =
-rqe,,)(e,,) +
$a4 +
(a6 + a3) co2e]v,,,
+Jp4 +
(as - a3)(302
e]v,,,
+:(a6 +
a3) sin(2+,,y +$0.5 -
a3) sin(2++,t,, = -p - qe,J2 - $f2 sin(2e)S + a(2 cyl c0s2 e + a2 + a3 + a5 + CQ) sin(28)v,,,
+:(~cY~ ~05~ 8 - a2 - a3 + ff5 + CV~) sin(2e)v,,,
+[a1 ~0s~ e + a4 + (a5 + CQ) ~0s~ e12),,, + $kl sin2(2e)v,,,.
In above equations, y andp are arbirary (indeterminate) constants, and y1 = a3-a2, y2 = a3 + a2 and al,...& are the viscosity coefficients following the notations of Ref. (14).
Combining Eq. (A&) and Eq. (A.2c) by subtracting Eq. (A.2a) x cod and Eq. (A.2c)
x sine, one can obtain: !-j&,nbn’,l sin(2e)
c[(n:)2 - E~ sin2 81312 + Iwe - -d + + - y2 cos(2e)~V,,,
1 (A-3)
-$71 + 72 coswlv,,, - y2 sin e c0s e& - v,,~) = 0.
where1 = ( <E2 >/4x) (l/m’,& and n’,~ = ,lo’,le’[(lIe1)2-E~sin 26) -*‘2 is the effective refractive index of the e-beam.
The equations are so complicate, it is impossible to solve these equations of motion analyti-cally. Even using numerical approach, it would still be a terrible tedious work. To simplify these
E!!!!!
VOL. 30 SHYU-MOU CHEN AND RU-PIN PAN 141
equations, we consider the case with small angle 13, therefore we can set sin6 2~ 8 and cos8 =L 1, then linearize these equations by keeping only terms up to the first order in 8 and vi. The results
are
c-‘IO + IiT28 - 7,e - cwp,,, - Q3V,,, = 0,
1
@3@),z + (a5 + Q+,* + s(“3 + a4 + as)V2v, = 0 ,
Icy2 + a3 + a5(1 - a3/a2)]V,,y,t + ff4v’Vy = 0, Q37Jz,y + (Yzvy,z -- 0, (A.4a) (A.4b) (A.4c) (A.4d) (A.4e) where C-t s n,‘[l-(n,‘/n,‘)2]/c, c is the speed of light, and we have used the incompressible con-dition Of material, i.e., CiVQ = 0.7
Physically, from Eq. (A.4a), we know that the external light field < E2 > , which is a func-tion of transverse coordinate,x andy, induces molecular reorientafunc-tion, 8, and 8 induces the fluid-flow (v,,v,,v,) through Eqs. (A.4b)-(A.4e). Conversely, v, and v, also influence 0 through Eq. (A.4a). The induced fluid-flow is the so-called backflow.6
Now comparing the force sources of the backflow, i.e., az(& in Eq. (A.4b) and a~(@,~ in Eq. (A.4c), because ] (8), 1 5 I(& 1 due to the non-local effect,’ and ]a3 1 < < Ia2 ( for Xl3 in the nematic phase, I5 1 a3(8), I is much less ( CQ(~),~ I. Therefore I v, ( is much smaller than
]vx] .
The effect of v, can be neglected in the Eqs. (A.4a) and (A.4b). Moreover, comparing the coefficients of vqz and vzJ in these two equations: Ia3 I -z c 1~x2 1 and I4(a5 + al) I I ) (as - ~22) 1, we can also conclude that the effect of v, can be neglected. The effect of vr can be also neglected by similar considerations through Eqs. (A.4d) and (A.4e). Thus the equations of mo-tion are further simplified as following:c-l10 + I{(@ + vp>e - 7rd, - cY’2v,,, = 0, 1
ff@),, + n,d,2v2 + 2Q40:v, = 0,
(ASa)
(ASb)
where 7,rc = 1/2(a5 + a4 - a& a, 2 s d”/Jz2, and 9,” is the Laplacian operator in the transverse coordinate.
When the external applied field < E2 > is a uniform field, Vt26 = 9,% = 0, and the Eqs. (ASa) and (A.5b) are reduced to the simple forms as for the case of the uniform magnetic field.”
.
142 ANALYSIS OF BEAM SIZE EFFECT OIi THE LASER INDUCED...
REFERENCES
VOL. 30
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