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Determination of liquid crystal parameters by the attenuated total reflection technique
Volume
118. number
PHYSICS
3
LETTERS
6 October
A
1986
DETERMINATION OF LIQUID CRYSTAL PARAMETERS BY THE ATTENUATED TOTAL REFLECTION TECHNIQUE K. EIDNER,
G. MAYER
and R. SCHUSTER
Sektion Physik der Karl-Marx-Uniuersiiiit Received
20 April 1984, revised manuscript
Leipzig, Linnbtrasse received
5, DDR-7010
30 July 1985; accepted
Leipzig,
GDR
for publication
23 July 1986
Theoretical considerations and measurements on a thin plane-parallel liquid crystal plate, surrounded by an optically dense medium, show that the fringes of equal inclination start at an angle of incidence less than the critical angle of total reflection. Despite its practical importance this well-known quantum effect was disregarded in optics up to now.
The attenuated total reflection (ATR) technique [l] is a well established experimental method for the measurement of liquid crystal (LC) properties such as the indices of refraction [2], the tilt angle of the optical axis in a sandwich cell [3] and, especially, the pretilt angle of the boundary LC layer [4-61 which is of great importance for the determination of the LC-to-wall anchoring anisotropy. Similar information can be obtained by measuring the transmittance instead of the reflectivity [7]. The interpretation of experimental data is usually based on Fresnel’s formulae which are valid for the reflection at a single interface, i.e. for semi-infinite media on both sides of the reflecting plane. However, experimental studies were mostly performed on thin sandwich cells with thicknesses of about 10 pm up to several hundreds of microns. Although the cell thickness is much larger than the wavelength X of the radiation used the penetration depth of the evanescent wave approaches the same order of magnitude near the critical angle. Hence, the presence of a second reflecting plane should affect the critical angle of total reflection of a LC layer of finite thickness. Furthermore, planes of total internal reflection [S] which necessarily occur in a distorted, initially homeotropic LC layer of positive optical anisotropy can act as a second reflecting plane [5]. In our experiments we used a room temperature nematic mixture doped with 0.2% by weight hexadecyltrimethylammoniumbr&nide to get a 0375-9601/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
uniform homeotropic alignment. The LC layer of 20 pm thickness was confined between two equal high-index glass prisms. A suitable mirror arrangement allowed one to direct a linear beam at this double prism cell which was placed in the slit of a magnet. The emergent light was guided by mirrors to a screen and could be observed or photographed, respectively. The experimental arrangement is depicted in fig. 1. Because of the rather large distance between the sample and the screen it was possible to resolve the fringes of
Fig. 1. Experimental arrangement. i, direction of the polarization of the incident laser beam (ordinary ray within the LC which undergoes a depolarized scattering). s, direction of the polarization of the emergent light (extraordinary ray within the LC). a,, angle of emergence of the m th fringe. @, angle of the direction of the magnetic field B to the cell normal. f?(z), r-dependent angle of the director to the cell normal.
B.V.
149
Volume 118, number 3
PHYSICS LETFERS A
equal inclination with the plane parallel LC plate up to the 17th interference order. We measured the angular positions of the interference fringes which occur in the reflected light as dark fringes on a bright background and in the transmitted and scattered light as bright fringes on an almost dark background. They were found to be equal for equal directions of polarization. Thus we restricted our observation to that scattered light which was excited by an ordinary polarized incident laser beam. The scattering process itself serves only as a diffuse illumination from the inner region of the sample. Because of absence of a directly reflected beam the interference patterns of the scattered light were equal on both sides of the double prism cell. By means of an external magnetic field we could induce distortions of varying strength in the LC sample. We used only such distortions where the local optical axes remain in the plane of incidence. In this case the T M and TE polarizations are decoupled. The TE waves are affected by the ordinary index of refraction whereas the TM waves are governed by a refractive index profile of the extraordinary index of refraction which depends on the director configuration [2-7]. When the magnetic field was increased we observed a shift of the whole interference pattern towards lower angles of emergence. Then, with increasing strength of the field, the first interference fringe fuses with the second and forms a single broader fringe. If the field was further increased the same happens with the 3rd and 4th fringe, the 5th and 6th and so on, where the number of fused fringes depends on the maximum possible distortion angle of the director• When the magnetic field is decreased the fused fringes split and the whole interference pattern returns to its initial position (see ref. [11]). The fused fringes do not appear in the transmitted light. In order to explain our observation we consider the optically uniaxial LC sample as a plane parallel plate with ordinary ( n • ) and maximum extraordinary (n11) refractive indices, surrounded by an isotropic medium (glass prism) with refractive index N ( > nil > n •). The reflection coefficient of such a plate is given by [9] r= 150
r12 + r23
exp(2ifl)
1 + r12r23 exp(2ifi) '
(1)
6 October 1986
where r12 and r23 are the reflection coefficients of the two interfaces and ]~ is the optical path between them in units of arc. In the case of an undistorted, uniformly aligned LC layer and for two equal media on both sides where r12 = - r 2 ~ we get from eq. (1) 1 - exp(2i/~) r = rl2 1 - r22 exp(2ifl)
(2)
For the optical path /~ we obtain 2 v n i nlld -
xg(0) [ g 2 ( ° ) -
2
,
(3)
with g2(0 )__n112 cos20+n±2 sin20, k, = N sin a, where 0 is the constant tilt angle of the director with respect to the normal of the plate and a is the angle of emergence within the prism [12]. Eq. (2) represents the oscillating behaviour of the reflectivity where the dark fringes in the reflected light are due to zeroes of I r [ 2. The interference fringes appear at angular positions which satisfy the condition 1~ = rn,~ where the integer m is the order of interference. We point out that for a fixed thickness d of the plate the first fringe (nearest to the critical angle) corresponds to the order of interference m = 1. The angular position where there would begin total reflection at a boundary between two semi-infinite media (1 and 2), i.e. the critical angle of total reflection, would correspond to the order of interference m = 0 (see below eq. (4)). This angle is neither accompanied by an interference fringe nor by another singularity nor by a fixed value of the reflectivity and therefore c a n n o t be d e t e c t e d . The reflection coefficient at this angle c~ = a~. is iim r =
~,~,
-ikld 2NZ/nlln • - ikld
k~ is the z-component (normal to the plate) of the wave vector in the outer medium. Although this is analogous to a well-known quantum effect (see ref. [10], problem 10), and eq. (2) implies this behaviour, it was disregarded in optics up to now
Volume 118, number 3
PHYSICS LETTERS A a) O- /
ofO
/
oJOJ
/oJ
°/2;/ -"
0,02
0,04
7/
0,06 [
.z
. z
S l n~lc-S m
"
a m
Fig. 2. D e p e n d e n c e of the interference o r d e r m on ( s i n 2 a c sin2a,,,)W2; (a) c o u n t of the orders started with m = 1 ; (b) c o u n t of the orders started w i t h m = 0.
(for instance ref. [9], p. 326). T h e e x p e r i m e n t a l evidence for this s t a t e m e n t is p r e s e n t e d in fig. 2. N a m e l y , rewriting eq. (3) we get for an u n d i s t o r t e d h o m e o t r o p i c ( O ( z ) = 0), L C layer
/3-
2 v n- -± N d ( s i n 2 a c - -
X nrl
• 2a ) ~1/2 ,
sin
(4)
with sin ac = nll/N. F r o m eq. (4) it can be seen easily that the o r d e r of interference m = / 3 / v should d e p e n d linearly on the q u a n t i t y ( s i n e a ~ - s i n 2 a m ) 1/2 with a zero intercept. But a p l o t of m = fl/~r versus (sinZac sinZam) 1/2 for an u n d i s t o r t e d s a m p l e leads to an i n t e r c e p t of m i n u s one (case (b) in fig. 2) when the first interference fringe is c o n s i d e r e d to be d u e to the o r d e r of interference m = 0, i.e. when the first fringe is t a k e n for the critical angle. A zero interc e p t as d e m a n d e d b y eq. (4) can be o b t a i n e d o n l y w h e n the first fringe is c o n s i d e r e d to be of the
6 October 1986
o r d e r of interference m = 1 a n d when a critical angle a c > a~ is d e t e r m i n e d b y e x t r a p o l a t i n g the linear d e p e n d e n c e of m 2 versus sin2am to m = 0 a c c o r d i n g to the square of eq. (4) (case (a) in fig. 2). Evidently, the interference p a t t e r n of the fringes of equal inclination with a p l a n e - p a r a l l e l plate of fixed thickness does not c o n t a i n any fringe m a r k ing the critical angle of total reflection. T h e systematic error caused b y a m i s i n t e r p r e t a tion of the first interference fringe as critical angle of total reflection increases with decreasing t h i c k ness. F o r a cell thickness of 10 ~tm it w o u l d be typically a b o u t 10 4 for the refractive indices a n d a b o u t l ° - 2 ° for the tilt angle of the director. The influence of d i s t o r t i o n s on the interference p a t t e r n m e n t i o n e d a b o v e will be discussed in a f o r t h c o m ing paper.
References [1] N.J. Harrick, Internal reflection spectroscopy (Interscience-Wiley, New York, 1967). [2] D. Rivi6re, Y. Levy and C. Imbert, Opt. Commun. 25 (1978) 206. [3] Y. Levy, D. Rivi~re, C. Imbert and M. Boise, Opt. Commun. 26 (1978) 225. [4] D. Rivirre, Y. Levy and E. Guyon, J. Phys. (Paris) Lett. 40 (1979) L-215. [5] D. Rivi~re and Y. Levy, Mol. Cryst. Liq. Cryst. (Lett.) 64 (1981) 177. [6] L. Komitov and A.G. Petrov, Phys. Stat. Sol. (a) 76 (1983) 137. [7] S. Naemura, Appl. Phys. Lett. 33 (1978) 1. [8] H.P. Hinov, Rev. Phys. Appl. 15 (1980) 1307. [9] M. Born and E. Wolf, Principles of optics (Pergamon, Oxford, 1975). [10] S. Flggge, Rechenmethoden der Quantentheorie (Springer, Berlin, .1976). [11] K. Eidner, G. Mayer and R. Schuster, Phys. Lett. A 118 (1980) 152. [12] G. Mayer, Dissertation, Leipzig (1984).
151
118. number
PHYSICS
3
LETTERS
6 October
A
1986
DETERMINATION OF LIQUID CRYSTAL PARAMETERS BY THE ATTENUATED TOTAL REFLECTION TECHNIQUE K. EIDNER,
G. MAYER
and R. SCHUSTER
Sektion Physik der Karl-Marx-Uniuersiiiit Received
20 April 1984, revised manuscript
Leipzig, Linnbtrasse received
5, DDR-7010
30 July 1985; accepted
Leipzig,
GDR
for publication
23 July 1986
Theoretical considerations and measurements on a thin plane-parallel liquid crystal plate, surrounded by an optically dense medium, show that the fringes of equal inclination start at an angle of incidence less than the critical angle of total reflection. Despite its practical importance this well-known quantum effect was disregarded in optics up to now.
The attenuated total reflection (ATR) technique [l] is a well established experimental method for the measurement of liquid crystal (LC) properties such as the indices of refraction [2], the tilt angle of the optical axis in a sandwich cell [3] and, especially, the pretilt angle of the boundary LC layer [4-61 which is of great importance for the determination of the LC-to-wall anchoring anisotropy. Similar information can be obtained by measuring the transmittance instead of the reflectivity [7]. The interpretation of experimental data is usually based on Fresnel’s formulae which are valid for the reflection at a single interface, i.e. for semi-infinite media on both sides of the reflecting plane. However, experimental studies were mostly performed on thin sandwich cells with thicknesses of about 10 pm up to several hundreds of microns. Although the cell thickness is much larger than the wavelength X of the radiation used the penetration depth of the evanescent wave approaches the same order of magnitude near the critical angle. Hence, the presence of a second reflecting plane should affect the critical angle of total reflection of a LC layer of finite thickness. Furthermore, planes of total internal reflection [S] which necessarily occur in a distorted, initially homeotropic LC layer of positive optical anisotropy can act as a second reflecting plane [5]. In our experiments we used a room temperature nematic mixture doped with 0.2% by weight hexadecyltrimethylammoniumbr&nide to get a 0375-9601/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
uniform homeotropic alignment. The LC layer of 20 pm thickness was confined between two equal high-index glass prisms. A suitable mirror arrangement allowed one to direct a linear beam at this double prism cell which was placed in the slit of a magnet. The emergent light was guided by mirrors to a screen and could be observed or photographed, respectively. The experimental arrangement is depicted in fig. 1. Because of the rather large distance between the sample and the screen it was possible to resolve the fringes of
Fig. 1. Experimental arrangement. i, direction of the polarization of the incident laser beam (ordinary ray within the LC which undergoes a depolarized scattering). s, direction of the polarization of the emergent light (extraordinary ray within the LC). a,, angle of emergence of the m th fringe. @, angle of the direction of the magnetic field B to the cell normal. f?(z), r-dependent angle of the director to the cell normal.
B.V.
149
Volume 118, number 3
PHYSICS LETFERS A
equal inclination with the plane parallel LC plate up to the 17th interference order. We measured the angular positions of the interference fringes which occur in the reflected light as dark fringes on a bright background and in the transmitted and scattered light as bright fringes on an almost dark background. They were found to be equal for equal directions of polarization. Thus we restricted our observation to that scattered light which was excited by an ordinary polarized incident laser beam. The scattering process itself serves only as a diffuse illumination from the inner region of the sample. Because of absence of a directly reflected beam the interference patterns of the scattered light were equal on both sides of the double prism cell. By means of an external magnetic field we could induce distortions of varying strength in the LC sample. We used only such distortions where the local optical axes remain in the plane of incidence. In this case the T M and TE polarizations are decoupled. The TE waves are affected by the ordinary index of refraction whereas the TM waves are governed by a refractive index profile of the extraordinary index of refraction which depends on the director configuration [2-7]. When the magnetic field was increased we observed a shift of the whole interference pattern towards lower angles of emergence. Then, with increasing strength of the field, the first interference fringe fuses with the second and forms a single broader fringe. If the field was further increased the same happens with the 3rd and 4th fringe, the 5th and 6th and so on, where the number of fused fringes depends on the maximum possible distortion angle of the director• When the magnetic field is decreased the fused fringes split and the whole interference pattern returns to its initial position (see ref. [11]). The fused fringes do not appear in the transmitted light. In order to explain our observation we consider the optically uniaxial LC sample as a plane parallel plate with ordinary ( n • ) and maximum extraordinary (n11) refractive indices, surrounded by an isotropic medium (glass prism) with refractive index N ( > nil > n •). The reflection coefficient of such a plate is given by [9] r= 150
r12 + r23
exp(2ifl)
1 + r12r23 exp(2ifi) '
(1)
6 October 1986
where r12 and r23 are the reflection coefficients of the two interfaces and ]~ is the optical path between them in units of arc. In the case of an undistorted, uniformly aligned LC layer and for two equal media on both sides where r12 = - r 2 ~ we get from eq. (1) 1 - exp(2i/~) r = rl2 1 - r22 exp(2ifl)
(2)
For the optical path /~ we obtain 2 v n i nlld -
xg(0) [ g 2 ( ° ) -
2
,
(3)
with g2(0 )__n112 cos20+n±2 sin20, k, = N sin a, where 0 is the constant tilt angle of the director with respect to the normal of the plate and a is the angle of emergence within the prism [12]. Eq. (2) represents the oscillating behaviour of the reflectivity where the dark fringes in the reflected light are due to zeroes of I r [ 2. The interference fringes appear at angular positions which satisfy the condition 1~ = rn,~ where the integer m is the order of interference. We point out that for a fixed thickness d of the plate the first fringe (nearest to the critical angle) corresponds to the order of interference m = 1. The angular position where there would begin total reflection at a boundary between two semi-infinite media (1 and 2), i.e. the critical angle of total reflection, would correspond to the order of interference m = 0 (see below eq. (4)). This angle is neither accompanied by an interference fringe nor by another singularity nor by a fixed value of the reflectivity and therefore c a n n o t be d e t e c t e d . The reflection coefficient at this angle c~ = a~. is iim r =
~,~,
-ikld 2NZ/nlln • - ikld
k~ is the z-component (normal to the plate) of the wave vector in the outer medium. Although this is analogous to a well-known quantum effect (see ref. [10], problem 10), and eq. (2) implies this behaviour, it was disregarded in optics up to now
Volume 118, number 3
PHYSICS LETTERS A a) O- /
ofO
/
oJOJ
/oJ
°/2;/ -"
0,02
0,04
7/
0,06 [
.z
. z
S l n~lc-S m
"
a m
Fig. 2. D e p e n d e n c e of the interference o r d e r m on ( s i n 2 a c sin2a,,,)W2; (a) c o u n t of the orders started with m = 1 ; (b) c o u n t of the orders started w i t h m = 0.
(for instance ref. [9], p. 326). T h e e x p e r i m e n t a l evidence for this s t a t e m e n t is p r e s e n t e d in fig. 2. N a m e l y , rewriting eq. (3) we get for an u n d i s t o r t e d h o m e o t r o p i c ( O ( z ) = 0), L C layer
/3-
2 v n- -± N d ( s i n 2 a c - -
X nrl
• 2a ) ~1/2 ,
sin
(4)
with sin ac = nll/N. F r o m eq. (4) it can be seen easily that the o r d e r of interference m = / 3 / v should d e p e n d linearly on the q u a n t i t y ( s i n e a ~ - s i n 2 a m ) 1/2 with a zero intercept. But a p l o t of m = fl/~r versus (sinZac sinZam) 1/2 for an u n d i s t o r t e d s a m p l e leads to an i n t e r c e p t of m i n u s one (case (b) in fig. 2) when the first interference fringe is c o n s i d e r e d to be d u e to the o r d e r of interference m = 0, i.e. when the first fringe is t a k e n for the critical angle. A zero interc e p t as d e m a n d e d b y eq. (4) can be o b t a i n e d o n l y w h e n the first fringe is c o n s i d e r e d to be of the
6 October 1986
o r d e r of interference m = 1 a n d when a critical angle a c > a~ is d e t e r m i n e d b y e x t r a p o l a t i n g the linear d e p e n d e n c e of m 2 versus sin2am to m = 0 a c c o r d i n g to the square of eq. (4) (case (a) in fig. 2). Evidently, the interference p a t t e r n of the fringes of equal inclination with a p l a n e - p a r a l l e l plate of fixed thickness does not c o n t a i n any fringe m a r k ing the critical angle of total reflection. T h e systematic error caused b y a m i s i n t e r p r e t a tion of the first interference fringe as critical angle of total reflection increases with decreasing t h i c k ness. F o r a cell thickness of 10 ~tm it w o u l d be typically a b o u t 10 4 for the refractive indices a n d a b o u t l ° - 2 ° for the tilt angle of the director. The influence of d i s t o r t i o n s on the interference p a t t e r n m e n t i o n e d a b o v e will be discussed in a f o r t h c o m ing paper.
References [1] N.J. Harrick, Internal reflection spectroscopy (Interscience-Wiley, New York, 1967). [2] D. Rivi6re, Y. Levy and C. Imbert, Opt. Commun. 25 (1978) 206. [3] Y. Levy, D. Rivi~re, C. Imbert and M. Boise, Opt. Commun. 26 (1978) 225. [4] D. Rivirre, Y. Levy and E. Guyon, J. Phys. (Paris) Lett. 40 (1979) L-215. [5] D. Rivi~re and Y. Levy, Mol. Cryst. Liq. Cryst. (Lett.) 64 (1981) 177. [6] L. Komitov and A.G. Petrov, Phys. Stat. Sol. (a) 76 (1983) 137. [7] S. Naemura, Appl. Phys. Lett. 33 (1978) 1. [8] H.P. Hinov, Rev. Phys. Appl. 15 (1980) 1307. [9] M. Born and E. Wolf, Principles of optics (Pergamon, Oxford, 1975). [10] S. Flggge, Rechenmethoden der Quantentheorie (Springer, Berlin, .1976). [11] K. Eidner, G. Mayer and R. Schuster, Phys. Lett. A 118 (1980) 152. [12] G. Mayer, Dissertation, Leipzig (1984).
151