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Langmuir l B B S , l l , 699-704
Fresnel Formula for Optically Anisotropic Langmuir Monolayers: An Application to Brewster Angle Microscopy Yuka Tabe* and Hiroshi Yokoyama Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305, Japan Received October 10, 1994. I n Final Form: December 20, 1994@ A n approximateyet highly accurate Fresnel formula, describing the reflection from optically anisotropic Langmuir monolayers,is derived in a simple closed form. The utility ofthe Fresnel formula is demonstrated by application-to Brewster angle microscopy.
Introduction The Brewster angle microscope (BAM) is a n excellent tool for observation of macroscopic structuresin Langmuir monolayers. It was developed by Hen6n and Meunier' and by Honig and Mobius2in 1991, based on the property of the Brewster angle that the reflection of p-polarized light from the bare air-water interface vanishes, thereby allowing high-contrast intact visualization of monomolecular films on water surface without the aid offluorescent probes. The BAM has since then spread widely in morphology studies of Langmuir monolayers, mostly associated with phase separation^.'-^ An advanced use of BAM employs a n analyzer, as in a polarizing microscope, which helps amplify the contrast from the optical anisotropy, ubiquitous in Langmuir monolayers. This technique, which we refer to a s the depolarized Brewster angle microscopy (DBAM), has recently been applied to monolayer analogs ofvarious types of smectic liquid crystals,8-11 exhibiting characteristic optical anisotropies. (With further insertion of a quarterwave plate, the DBAM may be more properly designated as an ellipsometric microscopy (EM).6~7)And it has successuflly identified intriguing in-plane orientational superstructures such a s the stripe domains comprised of x-walls in smectic C-like Langmuir monolayerslOJ1and the multiarmed star defects and the shell domains occurring in the hexatic smectic phases of Langmuir monolayers.12,13 Although the DBAM is functionally similar to the depolarized reflected light microscopy (DRLM), extensively used during the last decade for observing freely suspended smectic films,14-17the lack of compact Fresneltype formula in the case of DBAM, due to the strongly oblique incidence, has hampered straightforward interAbstract published in Advance ACS Abstracts, February 1, 1995. (1)H h o n , S.;Meunier, J, Rev. Sci. Instrum. 1991,62 (41,936. (2)H h i g , D.;Mobius, D. J.Phys. Chem. 1991,95,4590. (3)H h o n , S.; Meunier, J. Thin Solid Films 1992,210,121. (4)Honig, D.;Mobius, D. Thin Solid Films 1992,210,64. (5)Mul, M. N. G.; Mann, J. A.,Jr. Langmuir 1994,10,2311. (6)Reiter, R.; Motschmann, H.; Orendi, H.; Nemetz, A.; Knoll, W. Langmuir 1992,8,1784. (7)Paudler, M.; Ruths, J.; Riegler, H. Langmuir 1992,8,184. (8)Overbeck, G. A.;Honig, D.; Mobius, D. Langmuir l993,9,555. (9) . . Overbeck. G. A.: Mobius. D. J. Phvs. Chem. 1993.97. 7999. (10)Adams, J.; Rettig, W.; Duran, R."S.; Naciri, J.; Shashidhar, R. J.Phys. Chem. 1993,97,2021. (11)Tabe, Y.; Yokoyama, H. J . Phys. SOC.Jpn. 1994,63 (7),2472. (12)Ruiz-Garcia, J.;Qiu, X.; Tsao,M.; Marshall, G.; Knobler, C. M.; Overbeck, G. A.; Mobius, D. J . Phys. Chem. 1993,97,6955. (13)Overbeck, G. A,; IIonig, D.; Mobius, D. Thin Solid Films 1994, 242,213. (14)Pindak, R.; Young, C. Y.; Meyer, R. B.; Clark, N. A. Phys. Rev. Lett. 1980,45(14),1193. (15)Langer, S. A.; Sethna, J. P. Phys. Rev. A 1986,34 (61,5035. (16)Bahr, Ch.; Fliegner, D. Liq. Cryst. 1993,14 (2),573. (17)Maclennan, J.E.;Sohling, U.; Clark, N. A.; S e d , M. Phys. Rev. E 1994,49 (4),3207. @
pretation of DBAM images in relation to the molecular orientation in monolayers as compared to the case of DRLM. Previous studies of optically anisotropic monolayers, using not only DBAM but also ellipsometry, had to invariably employ numerical analyses of reflectivity to get information on the optical properties of monolaye r ~ . ~ , These ~ ~ ,numerical ~ ~ , ~analyses ~ , ~ ~ involve a number of basically unknown parameters such as the film thickness, dielectric constants, tilt, and azimuthal angles of the optic axis, so that it is always difficult to assess, with confidence, the immunity of final results to changes in these parameters, and more importantly to dig out, if any, a smaller number of controlling parameters through which the reflectivity behavior is essentially determined. In this paper, we derive a n approximate yet highly accurate Fresnel formula describing the reflection from optically uniaxial as well as biaxial Langmuir monolayers, on the basis of Berreman's 4 x 4 matrix formalism.20A particularly notable consequence of the Fresnel formula is that the qualitative features of the DBAM images of optically uniaxial monolayers are shown to be governed by a single parameter consisting of the tilt angle of the optic axis and the ordinary dielectric constant. In retrospect, the essential content of the present formula can be found even in rather old ellipsometry literature;21,22 however, it has not been given such an intuitively transparent form as to force later researchers to recognize it as S U C ~ . ~ J ~We J ~demonstrate J~J~ here the utility of our Fresnel formula, concentrating on its application to the DBAM in view of its practical significance; a brief account of ellipsometry is given in Appendix B.
Fresnel Formula for Langmuir Monolayers The 4 x 4 matrix formalism is a standard mathematical tool to solve the problem of light propagation through optically anisotropic layered materials whose properties vary only along the layer normal (hereafter taken as the z-axis). The form of Maxwell's equations that the 4 x 4 matrix formalism deals with is generally written as
V X H Z a-0 at
V X E = - -aB at
(18)Overbeck, G.A.;Honig, D.; Wolthaus, L.; Gnade, M.; Mobius, D. Thin Solid Films 1994,242,26. (19)Hosoi, IC;Ishikawa, T.;Tomioka, A.; Miyano, K. Jpn. J.Appl. Phys. 1993,32 (21,L135. (20)Berreman, D.W. J. Opt. SOC.Am. 1972,62 (4),502. (21) Dignam, M. J.; Moskovits, M.; Stobie, R. W. Trans. Faraday SOC.1971,67,3306. (22)Ayoub, G. T.;Bashara, N. M. J . Opt. SOC. A m . 1978,68(7),978.
0743-746319512411-0699$09.00/0 0 1995 American Chemical Society
Letters
700 Langmuir, Vol. 11, No. 3, 1995
With the plane of incidence taken to be the xz-plane, the differential operators for a monochromatic plane wave are reduced to
$)
V = (ikx,0,
a- -iw at
where k, is the x-component of the wave vector and cr) is the angular frequency. By combining eq 1and eq 2 and eliminating E, and H,, we obtain a set of first-order simultaneous differential equations for E,, Ey,H,, andH,, which can be expressed in a matrix form as
(3)
Z
IC
Figure 1. The definition of Eulerian angles, specifyrng the orientation of the principaloptic axis (6,q , 5)and the laboratory frame of axis ( x , y, z). Here, a is the azimuthal angle between the projection of 5 axis onto xy-plane and x-axis,/?is the tilt angle between the 5 axis and the z-axis, and y is the angle of
rotation about the 5 axis. where
(4)
and
T= Figure 2. Reflection and transmission of plane wave through
(y
- EYY
-1
+ -2 €YZ
EZZ
0
a Langmuirmonolayer spread on the water surface. The plane
0 k x Eyz --
of incidence is taken t o be the xz-plane and the origin of z-axis
is taken at the air-monolayer interface.
k
and
Here, c and k = u/c are the velocity and the wavenumber of light in vacuum and EC is the i j-componentof the relative dielectric tensor in the laboratory frame of axis, which is related to the principal dielectric constants by
€=R[;
3
We now restrict our attention to the reflection from a Langmuir monolayer, whose thickness s is much smaller than the wavelength. We assume that the optical property of the monolayer may be characterized by a dielectric constant which is uniform across the monolayer thickness. For Az (YZ(O) Yl‘(0))
(12)
As mentioned above, the electromagnetic field in an isotropic medium has only two independent components, say E, and Ey;therefore, eq 12offers a sufficient condition to solve for the reflectivity as well as the transmissivity. After tedious yet straightforward calculations (see Appendix A), we obtain the reflectivity in the following closed form
On the basis of eq 14,the p-to-s-reflectance, lrsPl2,can be written in the following general form
(
I = d 2h 2 c 3 - -€ 2
f
~2 (A sin 2a + B cos 2 a +
‘1)2
C sin a + D cos aI2 (15) where
1 A = -[1 2
+ (p - l)u21 sin23!,
1
-p
(l
+ cos2 p) cos 2 y
B = -pu cos /3 sin 2 y
C = -q(l + u cos 27) cos p sin 8, D = -qu sinp sin 2 y where Ell and E l are the components of the electric field, parallel and perpendicular to the plane of incidence, respectively. The components of the reflectivity tensor are specifically given by
‘pp
rps= id
+ +
-n cos Bi cos 8, - n cos e, cos et 2
(n COS 8,
COS
H=
1
p cos2 p
+ [I + (1- p ) u cos 2y1 sin2p
and the parameters p , u , q , and h are defined by
+ idrfpp
e, cos e,
+ COS e,)(nCOS 8, + COS e,)
P=X
e, COS e, rsp= id X ( n cos 8, + COS e,)(nCOS e, + COS e,) COS
+ +
-n COS 9, COS ei ‘ ~ 5 = n cos e, cos ei
+ idr’ss
h=
E2
+ E1
2c3 - c2 - c1 2n2 tan et E2
2
(n COS 8,
263
€2 - €1
u=
4=
2
(16)
+
COS
COS
+ E1
eiCOS e,
e,)(nCOS 8,
+
COS
e,)
(17)
We have taken €3 as such a principal dielectric constant that gives the largest uniaxial dielectric anisotropy 163 (€2 ~1)/21. Hence, the parameter u has the meaning of the ratio between the biaxial and the uniaxial dielectric anisotropies, and by the choice of €3, we have IuI < V 2 . Uniaxial Monolayers. For optically uniaxial films with = €2, which would be a reasonable approximation for many known 2D liquid crystalline monolayers, the DBAM reflectivity takes a particularly simple form as
+
(14)
where Oi, et, and n are the incident angle, the angle of refraction in the subphase, and the refractive index of the subphase; for r f p pand rfss,see Appendix A. Here again, we have neglected terms of the order of d2 or higher in eq 14,since s is normally on the order of 1nm, so that d This is the Fresnel formula we are after. The higher order correction to eq 14 is roughly given by a quantity relative to the lower order contribuof the form 10-1d2~2 tions; here, E denotes a typical value of the monolayer dielectric constant. Since d % 0.03 for films with the thickness s x 3 nm (at 2 = 633 nm), the relative correction does not exceed 10-462, which amounts only to 1% even if E = 10.
-
Application to the DBAM In the DBAM configuration, each component of the reflectivity tensor can in principle be individually attained (except for the phase) by an appropriate arrangement of the polarizer and the analyzer. Of the two physically equivalent depolarized components, rspand rps,the former is experimentally more tractable, because the reflectivity of the s-polarized light is always bigger than that of p-polarized light, and this is especially so near the Brewster angle where the latter disappears and brings about the highest signal-to-noise ratio as in the BAM. Therefore, we concentrate here on rsp,and investigate its dependence on the relevant parameters.
I(a)= hq sin2a(cos a - f 1 2
(18)
where
and
A remarkable feature ofthis result is that the dependence of reflectivity on the azimuthal angle a is completely determined by a single parameter f,except for a constant factor. In particular, in the case of normal incidence and hence f = 0, the above formula reduces to that for a conventional polarizing microscope and DRLM as intuitively expected. For oblique incidence, however, the parameter f runs from 0 to infinity as the tilt angle of the optic axis moves from parallel to perpendicular relative to the air-water interface, i.e., p = n/2 0.
-
Letters
702 Langmuir, Vol. 11, No. 3, 1995 -1
-
-c:
fi
L....I........I
........I
: :
:
I : I A
........I....... .........I ........:.......I ....... ................:.......I............... ........ ....
t 0
270
180
90
Azimuthal angle
a
360
(degree)
\
0
180
90
270
t
360
a (degree) Figure 3. Normalized reflectivityas a functionofthe azimuthal angle a for opticallyuniaxial monolayers: (a, top)f = 2 > 1 and (b, bottom)f = 0.2 1. Azimuthal angle
f Incident direction
f
/ Figure 4. In-plane orientation of optic axis for an s = 1point defect.
The existence of nonzero f brings about a distinctive difference on the reflectivity behavior of DBAM as a function of the azimuthal angle from that of the polarizing microscope and DRLM. For a 2n rotation of a,I(a)exhibits two or four intensity minima and maxima as f > 1or f .e 1, as shown in Figure 3. Under the DBAM, therefore, as s = 1 point defect, frequently found in smectic-C-like monolayers,lOJ1could manifest two different types of textures having two or four dark brushes as illustrated in Figure 4. Specifically, when f > 1, there appear two dark brushes with different widths at a = 0 and a = z, together with two bright bands of equal intensity at a = cos-l[(f - (f2 8)1/2)/4],as shown in Figure 5a. On the other hand, when f < 1, there appear four dark brushes with different widths emanating from the point defect at a = 0, n,c o s 1 f, and four bright bands with differing intensities at a = cos-l[(ff (f2 8)lI2/4]as shown in Figure 5b. These features are quite different from that of polarizing microscopy and DRLM, which always exhibit four dark brushes with equal width at a = 0, d 2 , n,3 d 2 and four bright bands with equal intensity at a = d 4 , 3n/4,5n/4, 7n/4. This 4-fold degeneracy is broken in the DBAM by the oblique incidence. These properties of DBAM images make it possible to uniquely determine a; the center of the thinner dark brush located between the two brightest bands corresponds to a = n for any value off.
+
+
Figure 5. DBAM images of an s = 1point defect configuration in a uniaxial monolayer (seeFigure 4). The center of the image corresponds to the point defect and the images are invariant to changes in the spatial scale. The arrow shows the incident direction. (a, top) f = 2 > 1 and (b, bottom) f = 0.2 < 1.
Since the parameter f contains the tilt angle p, a quantitative analysis of DBAM reflectivity also gives information about the tilt angle, as attempted via numerical calculations by Hosoi et al.19 For example, in the presence of an s = 1point defect, an estimate off can be directly gained from the position of the brightest band, aM, by the equations, f = 2 cos a M - (l/cos a d , and hence we get €1 tan p = [cos a ~ / (cos2 2 a M - l)ln2tan et. Even in case s = 1 point defects are not available, the same principle can still apply, if the plane of incidence is rotated with regard to the monolayer.18 Influence of Biaxiality. For optically biaxial films (€1 f ea), the reflectivity cannot in general be given in as simple a form as for uniaxial monolayers, since B and D do not disappear in eq 15 except at y = 0 or n/2. The consequence of the finite B and D terms is to make I(a) asymmetric for a -a. To study the reflectivity as a function of a in detail, it is convenient to rewrite eq 15 as
-
where
Jc2+ o2 K=
JAZ3
Langmuir, Vol. 11, No. 3, 1995 703
Letters
a' = a
h2 = 6h(r3 -
+ tan-'($)
9) H
J
m
(22)
From this equation, we see that the number of dark brushes and bright bands during 2n rotation of a is determined by two parameters K and Q. In contrast to the uniaxial case, which allows only two or four brushes, it can be readily shown that two, three, or four brushes can now appear. Their actual positions with respect to the plane of incidence, however, depends further on the phase tan-l (DIG). For s = 1point defect, an example of three-band texture in shown in Figure 6. Figure 7 shows the boundaries on the K - Q plane separating the two, three and four band regions. As mentioned above, all the positions of the dark and bright bands are variable, and I(a>does not necessarily vanish a t a = 0, n.
Figure 6. DBAM image for an s = 1 point defect configuration in a biaxial monolayer (see Figure 4): K = 1.3, Q = -3/4n.The arrow shows the incident direction.
On the other hand, when y = 0 or n/2 a t which point the minor principal axes lie parallel and perpendicular to the interface, the reflectivity assumes exactly the same form as in the uniaxial case provided the parameter f i n eq 19 is replaced by: for y = 0
f=
l + V
+
1 (p - l)v2 -
+
1 cos2 p PV sin2P
n2tan 0, (23) €1 tan P
and for y = n/2
f=
1-V
+
1 (p - l)v2
cos2 p + 1 +sin2 p PV
n2tan 0, (24) €1 tan P
0
1
2
3
5
K Figure 7. Parameter regions on the K-Q plane, giving two (11), three (III), and four (W) dark brushes for an s = 1 point defect in an optically biaxial monolayer under the DBAM.
Appendix A. Derivation of Reflectivity. Equation 12 can be written as
-&(E: 1 Therefore, the DBAM images for this case are indistinguishable from those for uniaxial films only on qualitative ground. One should note, however, that, since the P-dependence off and I differs between the uniaxial and the biaxial cases, the same texture could result in a different estimate of the tilt angle, depending on the presumed symmetry of the monolayer.
4
cos
ei
- EL)
'(A. 1)
When E: and El are eliminated, the above equation is rewritten as shown in eqA.2, where P i is the i j-component of the propagation matrix P. Because 6 is of the order of components of the reflectivity matrix may be well approximated by an expansion up to first order in 6,
Letters
704 Langmuir, Vol. 11, No. 3, 1995 resulting in eqs 13 and 14. And fPp and rf,, in eq 14 are written out as follows:
and
2n2 sin e, tan e, (n2- tan2 e,)(i - n2) rfss=
2 COS ei
(n COS
e, + COS
-n2 cos2 e,
-
sin2e, Eyy
n2-3+
+
E,,
r tan qe'* = PP
)
+
-5)(A.3)
B. Ellipsometry Formulas. For characterization of thin films by ellipsometry, it is customarily assumed that the film is nonabsorbing and uniaxial with the optic axis along layer n 0 1 ~ n a 1 . ~ In~this ~ ~ case, , ~ ~ the ellipsometric parameters, A and y j , are related to the reflection coefficients by
+
2(n2 1) n2 2(n2 1) 2 3sin 8, - 7 2n2 (B.5) n2 sin 8,
In particular, when the film thickness is sufficiently small to satisfy 6
