THE JOURNAL OF CHEMICAL PHYSICS 122, 174901 共2005兲
Propagation of an electromagnetic wave in an absorbing anisotropic medium and infrared transmission spectroscopy of liquid crystals B. K. P. Scaife and J. K. Vija兲 Laboratory of Advanced Materials, Department of Electronic and Electrical Engineering, Trinity College, University of Dublin, Dublin 2, Ireland
共Received 29 November 2004; accepted 26 January 2005; published online 2 May 2005兲 The theory of absorbance is developed for the entire electromagnetic spectrum of radiation in a semi-infinite anisotropic medium with a second rank dielectric tensor, the elements of which are complex and frequency dependent. The theory of the absorbance A共 , 兲 of an optically anisotropic liquid in an infrared 共IR兲 test cell is then outlined and applied to IR transmission experiments. A formula for the dependence of A共 , 兲, on 共 being the angle between the electric vector and the principal optical axis兲 is derived from first principles. The formula, for radiation of angular frequency , viz, A共 , 兲 = −log10关10−A共,0兲cos2 + 10−A共,/2兲sin2兴 is in agreement with that proposed by Jang, Park, Maclennan, Kim, and Clark 关Ferroelectrics 180, 213 共1996兲 兴 and confirms some of the work of Kocot, Wrzalik, and Vij 关Liq. Cryst. 21, 147 共1996兲兴. The comments on this formula by Jang, Park, Kim, Glaser, and Clark 关Phys. Rev. E 62, 5027 共2000兲兴, and by Kocot et al. are discussed. The absorbance A共 , 0兲 and A共 , / 2兲 have been expressed in terms of the optical properties of the material and the dimensions of the cell. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1874833兴 Iin共兲 = Iref共兲.
I. INTRODUCTION
Infrared absorbance spectroscopy is a powerful technique for studying the molecular structure and this technique has also been applied to the study of liquid crystalline material1 in the nematic phase since the early 1970’s. This technique was first applied in the early 1990’s to the study of the ferroelectric liquid crystalline materials and in particular for determining the tilt angle and the orientational order parameter.2 Recently, the technique has been used to determine a complete set of second rank orientational order parameters.3,4 The orientational distribution function of the directors in smectic layers has been determined for the ferroelectric liquid crystals first by Fukuda et al.5 In this paper we theoretically investigate the absorbance as a function of the angle of polarization where a ferroelectric liquid crystal is confined to lie between the two windows of a cell. The measurement of the absorption of infrared 共IR兲 radiation in a liquid crystal involves an arrangement shown schematically in Fig. 1. The radiation is divided by a beam splitter A; one part constituting a reference beam of intensity Iref, the other enters the test cell B with intensity Iin. The transmittance of the test cell, Tcell共兲, for radiation with angular frequency , is defined by the relation, Iout共兲 , Tcell共兲 ⬅ Iin共兲
共1.1兲
in which Iout共兲 is the intensity of the beam leaving the test cell and Iin共兲 is the intensity of the beam entering the test cell. We shall assume that
共1.2兲
Suppose that both these beams have the same crosssectional area then we may express Tcell共兲 in terms of the corresponding power-flux densities, Pref共兲 and Pout共兲, as follows: Tcell共兲 =
Iout共兲 Pout共兲 Pout共兲 = = . Iref共兲 Pref共兲 Pref共兲
共1.3兲
The units of Pref共兲 and Pout共兲 are W m−2. If the radiation is plane polarized and if the test cell contains an optically anisotropic substance, the power-flux density Pout, and hence the transmittance Tcell, will depend on , the angle between the electric vector of the incident beam and the z 共or 3兲 axis of the test cell. See Fig. 2. Therefore we replace Eq. 共1.3兲 by Tcell共, 兲 =
Pout共, 兲 . Pref共, 兲
共1.4兲
In Sec. III we shall show, in Eq. 共3.14兲, that in general, Tcell共, 兲 = ⌳储共兲cos2 + ⌳⬜共兲sin2 ,
共1.5兲
the factors ⌳储共兲 and ⌳⬜共兲 depend upon the dimensions and optical properties of the test cell and on the optical properties of its contents. Unfortunately there are no simple, general expressions for ⌳储共兲 and ⌳⬜共兲. The absorbance of the test cell, A共 , 兲, is defined by the equation, A共, 兲 ⬅ − log10Tcell共, 兲,
共1.6兲
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2005/122共17兲/174901/11/$22.50
this result with Eq. 共1.5兲 leads to
122, 174901-1
© 2005 American Institute of Physics
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174901-2
J. Chem. Phys. 122, 174901 共2005兲
B. K. P. Scaife and J. K. Vij
FIG. 1. The IR radiation is divided by the beam splitter A. The reflected portion is used as a reference; the transmitted portion enters the test cell B.
10−A共,兲 = Tcell共, 兲 = ⌳储共兲cos2 + ⌳⬜共兲sin2
共1.7兲
and hence 10−A共,0兲 = ⌳储共兲,
共1.8a兲
10−A共,/2兲 = ⌳⬜共兲.
共1.8b兲
With the following notation: A共,0兲 = A储共兲,
共1.9a兲
A共, /2兲 = A⬜共兲,
共1.9b兲
Eq. 共1.6兲 may be written in the form A共, 兲 = − log10关10−A储共兲cos2 + 10−A⬜共兲sin2兴. 共1.10兲 6
This equation was proposed by Jang et al. who obtained values of A储共兲 and A⬜共兲 by fitting it to experimental data. Several materials have been examined by means of Eq. 共1.10兲 such as the helical structure of a SmC* phase as well as the state which is fully unwound by an electric field.7–12 The measurements were made on liquid crystalline samples of thicknesses of up to 15 µm. The fully unwound SmC* state is known to have a finite degree of biaxiality. Jang et al. state in one of their recent papers10 关as a footnote in their reference 13兴, without providing reasons, that Eq. 共1.10兲 is exact only in the absence of birefringence 共no anisotropy of the dielectric permittivity兲. Because in most experiments the IR radiation is polarized nearly along the principal optical axis of the dielectric, Jang et al.10 assume that the birefringence may be ignored in samples of a few micrometers in thickness. From our derivation of Eq. 共1.7兲 in Sec. III 关see Eq. 共3.14兲兴 it does not seem that it is subject to such limitations. To clarify the discussion we outline, in Sec. II A, the theory of the propagation of electromagnetic waves in anisotropic media and in Sec. II B we discuss the passage of an IR beam through a test cell containing an anisotropic medium. Expressions for the Poynting vector are obtained in Sec. III, and those for the density of power absorption in Sec. IV. The molecular aspects of the problem are considered in Sec. V. A general discussion and conclusions will be found in Sec. VI, together with an analysis of the calculations of Kocot et al.7 It is found that, for a general molecular orientational distri-
FIG. 2. The arrangement for making IR absorption measurements. The radiation of wave vector k propagates along the y, or 2, axis. The sample lies between two 共a , b兲 CaF2 windows, each with an indium tin oxide 共ITO兲 electrode 共c , d兲. The smectic layers 共f兲 lie in the x-y plane. The electric vector E0 of the polarized radiation is perpendicular to k and makes an angle with the z axis which is taken to be the principal optical axis.
bution function, the equation derived by Kocot et al.7 is obtained in the absence of quadrupolar order 关D = 0 in Eq. 共6.9兲兴; this equation is also valid in the case of thin samples with C = D = 0.
II. MACROSCOPIC CONSIDERATIONS A. Homogeneous plane-wave propagation in a semi-infinite anisotropic medium
In this section we derive expressions for the electric field of a polarized, homogeneous plane-wave launched into the infinite plane boundary of a semi-infinite optically anisotropic medium. A homogeneous wave is one in which the planes of constant amplitude and those of constant phase are parallel to one another. The ensuing calculations are based on the treatment by Landau and Lifshitz.13 A more general treatment on the propagation of the electromagnetic wave in an infinite medium has been given by Yuan et al.14,15 in which they also consider spatial dispersion, i.e., permittivity to be spatially dependent on the wave vector, in addition to its frequency dependence. They expand the dielectric tensor as a Taylor series expansion to first order in the wave vector k. However, in the analysis given here, we confine ourselves to investigating the propagation in an absorbing anisotropic medium confined to a cell and ignore spatial dispersion. We can justify this by stating that the wavelengths considered here are too large compared with the intermolecular separations in the liquid crystalline material and furthermore this linear term in k is likely to introduce only a small correction. By explicitly including just the linear term in k, the calculations become too complicated to be helpful in the problem under discussion. We thus assume that all values of k are allowed for the different frequencies under discussion. For an insulating medium devoid of free electric charges, Maxwell’s equations 共in SI units兲 take the form
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174901-3
J. Chem. Phys. 122, 174901 共2005兲
Infrared transmission spectroscopy of liquid crystals
− k共兲 ⫻ H0共兲 = D0共兲,
共2.7兲
k共兲 · D0共兲 = 0,
共2.8兲
k共兲 · B0共兲 = k共兲 · H0共兲 = 0.
共2.9兲
Equations 共2.8兲 and 共2.9兲 indicate that k共兲 is perpendicular to D0共兲 and to H0共兲. When we combine Eqs. 共2.6兲 and 共2.7兲 we obtain the relation k共兲 ⫻ 关k共兲 ⫻ E0共兲兴 = − 20D0共兲. FIG. 3. The right-handed coordinate system with unit vectors i1 , i2, and i3.
curl h共r,t兲 = td共r,t兲,
By means of the standard formula from vector algebra, A ⫻ 共A ⫻ B兲 = A共A · B兲 − A2B,
curl e共r,t兲 = − tb共r,t兲, 共2.1兲
k共兲†k共兲 · E0共兲‡ − 关k共兲兴2E0共兲 = − 20D0共兲. 共2.12兲
in which the vectors e共r , t兲 , h共r , t兲 , d共r , t兲, and b共r , t兲 are, respectively, the electric field, the magnetizing force, the electric flux density, and the magnetic flux density, at position r at time t. The vector r has Cartesian components r1 , r2, and r3, such that 共2.2兲
in which i1 , i2, and i3 共see Fig. 3兲 are three mutually perpendicular unit vectors of a right-handed coordinate system. For a plane wave of angular frequency and complex, frequency-dependent propagation vector k共兲, the vectors e , h , d, and b may be expressed as follows: e共r,t兲 = Re E0共兲exp关ı共t − k · r兲兴,
共2.3兲
共2.13兲 in which summation over the repeated suffix  is implied in the last term. Thus Eq. 共2.13兲, written in full, reads as follows:
+ 1,3共兲E03共兲兴,
共2.14兲
D03共兲 = 0关3,1共兲E01共兲 + 3,2共兲E02共兲 + 3,3共兲E03共兲兴,
b共r,t兲 = 0h共r,t兲, 共2.4兲
in which 0 is the permeability of free space. The quantities E0共兲 , D0共兲, and H0共兲 are the complex vector amplitudes of e共r , t兲 , d共r , t兲, and h共r , t兲, at r = 0 and t = 0, for a particular propagation vector k共兲. This vector may be split into its real and imaginary components, and in keeping with the convention that counter-clockwise rotations are positive, we write 共2.5兲
To ensure that the plane wave is homogeneous, the vectors k⬘共兲 and k⬙共兲 must be parallel. The insertion of Eqs. 共2.3兲 and 共2.4兲 into Eqs. 共2.1兲 leads to − k共兲 ⫻ E0共兲 = − 0H0共兲,
兺 ␣,共兲E0共兲 = 0␣,共兲E0共兲,
=1
+ 2,3共兲E03共兲兴,
where ı = 冑−1, and Re denotes “real part of”. We shall restrict attention to nonmagnetic media, so that
k共兲 = k⬘共兲 − k⬙共兲.
3
D 0␣共 兲 = 0
D02共兲 = 0关2,1共兲E01共兲 + 2,2共兲E02共兲
b共r,t兲 = Re B0共兲exp关ı共t − k · r兲兴,
B0共兲 = 0H0共兲,
It is clear from this equation that D0共兲 need not be parallel to E0共兲. We cope with this possibility by introducing the complex, frequency-dependent, relative permittivity tensor ␣,共兲. The suffixes ␣ and  can take the values 1, 2, or 3. We therefore write
D01共兲 = 0关1,1共兲E01共兲 + 1,2共兲E02共兲
h共r,t兲 = Re H0共兲exp关ı共t − k · r兲兴, d共r,t兲 = Re D0共兲exp关ı共t − k · r兲兴,
共2.11兲
we may rearrange Eq. 共2.10兲 to read
div d共r,t兲 = div b共r,t兲 = 0,
r = i1r1 + i2r2 + i3r3 = i1x + i2y + i3z,
共2.10兲
共2.6兲
where 0 is the permittivity of free space. For the materials considered here, the permittivity tensor is symmetrical, that is, ␣,共兲 = ,␣共兲.
共2.15兲
Equation 共2.12兲, when expressed in tensor form is c20关k␣共兲k共兲E0 − k共兲k共兲E0␣共兲兴 = − ␣,共兲E0共兲,
共2.16兲
in which the summation convention is implied and where c0 = 冑共00兲−1 is the speed of light in free space. In view of the symmetry condition in Eq. 共2.15兲, it is always possible to choose a set of coordinate axes so that the permittiivity tensor becomes diagonal. Therefore, in matrix form, we have
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174901-4
J. Chem. Phys. 122, 174901 共2005兲
B. K. P. Scaife and J. K. Vij
关␣,共兲兴 =
冤
1共 兲
0
0
0
2共 兲
0
0
0
3共 兲
冥
共2.17兲
,
where we have used the abbreviations: 1,1共兲 = 1共兲, 2,2共兲 = 2共兲, and 3,3共兲 = 3共兲. With this result we may express Eq. 共2.16兲 in matrix form as follows: c20
冤
k201 − 共k22 + k23兲
k 1k 2 k202
k 1k 2 k 1k 3
−
共k21
k 1k 3 k23兲
+
k 2k 3 k203 − 共k21 + k22兲
k 2k 3
冤 冥
E01共兲 ⫻ E02共兲 = 0, E03共兲
冥
共2.18兲
FIG. 4. To illustrate the two possible transverse waves propagating along the 2 axis with propagation vectors k储 and k⬜.
k22k−2 0 = =
where we have introduced k0, the free-space wave number of the radiation, defined by the relation =
2 , k0 ⬅ = c0 0
共2.19兲
0 being the free-space wavelength. To save space in Eq. 共2.18兲 we have dropped the dependence on of k1 , k2, and k3. The condition that the matrix Eq. 共2.18兲 has a solution is that the determinant of the left-hand matrix should vanish, that is,
冨
k201 − k22 − k23
k 1k 2 k202
k 1k 2 k 1k 3
−
k21
k 1k 3 −
k23
k 2k 3
k 2k 3 k203 − k21 − k22
冨
= 0.
2 2 −4 2 2 −4 − 3共1 + 2兲k23k−2 0 + 1k 1k k 0 + 2k 2k k 0
共2.21兲
共2.22兲
and k ⬅ 关k共兲兴 = 兩k共兲兩
共2.25兲
These two solutions correspond, respectively, to plane waves with their electric vectors parallel either to the 1 共or x 兲 axis 关E⬜共兲 = i1E⬜共兲兴 or to the 3 共or z 兲 axis 关E储共兲 = i3E储共兲兴, see Fig. 4. If the medium is dissipative, the quantities 1共兲 and 3共兲 will be complex and, hence, so also will be the values of k2共兲 obtained from Eq. 共2.25兲. We shall use the following notation:
k储 = k储共兲 = i2k储共兲 = i2关k⬘储 共兲 − k⬙储 共兲兴,
2 关1⬘共兲 − 1⬙共兲兴, c20
2 关3⬘共兲 − ⬙3共兲兴. c20
Now we restrict our attention to a plane-wave traveling along the 2 共or y 兲 axis, so that k1共兲 = k3共兲 = 0. In this case Eq. 共2.21兲 reduces to the following quadratic in k22k−2 0 : 1共兲2共兲3共兲 − 2共兲关1共兲 + 3共兲兴k22k−2 0 +
2共兲k42k−4 0
= 0.
共2.24兲
On dividing throughout by 2共兲, we find the solutions of this equation to be
共2.27b兲
After separating the real and imaginary parts of these expressions, we find that
= k共兲 · k共兲 = 关k1共兲兴2 + 关k2共兲兴2 + 关k3共兲兴2 共2.23兲
共2.27a兲
k2储 = 关k储共兲兴2 = 关k⬘储 共兲 − k⬘储 共兲兴2
⬘ 共兲兴2 − 关k⬜ ⬙ 共兲兴2 = 关k⬜
2
= k21 + k22 + k23 .
共2.26兲
so that
= k203共兲 =
where we have used the facts that
2
1共 兲 . 3共 兲
= k201共兲 =
2 −2 123 − 1共2 + 3兲k21k−2 0 − 2共1 + 3兲k2k0
2
再 冎
2 ⬘ 共兲 − k⬜ ⬙ 共兲兴2 k⬜ = 关k⬜共兲兴2 = 关k⬜
After considerable algebra this equation reduces to
k共兲 = i1k1共兲 + i2k2共兲 + i3k3共兲
关1共兲 + 3共兲兴 ± 冑关1共兲 − 3共兲兴2 2
⬘ 共兲 − k⬜ ⬙ 共兲兴, k⬜ = k⬜共兲 = i2k⬜共兲 = i2关k⬜
共2.20兲
+ 3k23k2k−4 0 = 0,
关1共兲 + 3共兲兴 ± 冑关1共兲 + 3共兲兴2 − 41共兲3共兲 2
⬘ 共兲k⬜ ⬙ 共兲 = 2k⬜
2 ⬙1共兲, c20
关k⬘储 共兲兴2 − 关k⬙储 共兲兴2 = 2k⬘储 共兲k⬙储 共兲 =
2 1⬘共兲, c20
2 3⬘共兲, c20
2 3⬙共兲. c20
共2.28a兲
共2.28b兲
共2.28c兲
共2.28d兲
Thus, in view of these results and Eqs. 共2.1兲 and 共2.3兲, we have
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174901-5
J. Chem. Phys. 122, 174901 共2005兲
Infrared transmission spectroscopy of liquid crystals
k⬜共兲 ⫻ E⬜共兲 = i2 ⫻ i1k⬜共兲E⬜共兲
h储共y,t兲 = i1exp关− k⬘储 共兲y兴
= − i3k⬜共兲E⬜共兲 = − i30H⬜共兲 = 0H⬜共兲, 共2.29a兲 k储共兲 ⫻ E储共兲 = i2 ⫻ i3k储共兲E储共兲 = i10H储共兲 = 0H储共兲.
共2.29b兲
Since div b共r , t兲 = div h共r , t兲 = 0 everywhere, the vectors k and H must be perpendicular to one another. Consequently the set of vectors k⬜ , E⬜, and H⬜ and the set k储 , E储, and H储 are each mutually orthogonal. Therefore the vector pairs 共E⬜ , H⬜兲 and 共E储 , H储兲 both lie in the 1-3 plane perpendicular to the direction of propagation as in Fig. 4. We have the following relations between the magnitudes E⬜共兲, and H⬜共兲, and between the magnitudes E储共兲 and H储共兲: k ⬜共 兲 k ⬜共 兲 兩E⬜共兲兩 = E⬜共兲, 0 0 共2.30a兲 H储共兲 = 兩H储共兲兩 =
k储共兲 k储共兲 兩E储共兲兩 = E储共兲. 0 0
ph⬜共兲 =
ph储共兲 =
共2.31a兲
n1共兲 ⬅ n1⬘共兲 − n⬙1共兲 = 冑1共兲,
共2.34a兲
n3共兲 ⬅ n3⬘共兲 − n3⬙共兲 = 冑3共兲,
共2.34b兲
共2.35兲
兩共兲兩 = 冑关⬘共兲兴2 + 关⬙共兲兴2
共2.36兲
␦ = arctan关⬙共兲/⬘共兲兴
共2.37兲
and
then n共兲 = 冑兩共兲兩exp共− ı␦/2兲 = 兩n共兲兩关cos共␦/2兲 − ı sin共␦/2兲兴.
共2.31c兲
共2.38兲
关n⬘共兲兴2 − 关n⬙共兲兴2 = ⬘共兲
共2.39a兲
2n⬘共兲n⬙共兲 = ⬙共兲.
共2.39b兲
and
k储共兲 E储共兲 h储共y,t兲 = i1exp关− k⬙储 共兲y兴Re 0 ⫻exp兵关t − k⬘储 共兲y兴其.
共2.33b兲
Because, for a nonmagnetic medium, ␦ must lie somewhere in the range from 0 to it follows that n⬘共兲 can never be negative. Corresponding to Eqs. 共2.28兲 we have
k 共兲 ⬙ 共兲y兴Re ⬜ E⬜共兲 h⬜共y,t兲 = − i3exp关− k⬜ 0
⬘ 共兲y兴其, ⫻exp兵关t − k⬜
c0 c0 = = . k⬘储 共兲 Re冑3共兲 n⬘3共兲
共2.33a兲
with
共2.31b兲 and for the magnetizing force vectors:
c0 c0 = = , ⬘ 共兲 Re冑1共兲 n⬘1共兲 k⬜
共兲 = 兩共兲兩exp共− ı␦兲
e储共y,t兲 = i3exp关− k⬙储 共兲y兴Re E储共兲exp兵关t − k⬘储 共兲y兴其
共2.31d兲
If E⬜共兲 and E储共兲 are real quantities, these equations take the simpler forms
⬙ 共兲y兴cos关t − k⬜ ⬘ 共兲y兴E⬜共兲, e⬜共y,t兲 = i1exp关− k⬜ 共2.32a兲 e储共y,t兲 = i3exp关− k⬙储 共兲y兴cos关t − k⬘储 共兲y兴E储共兲, 共2.32b兲
+
共2.32d兲
in which the positive root is to be taken for positive traveling waves. If we express the complex, relative permittivity in the form
⬙ 共兲y兴Re E⬜共兲exp兵关t − k⬜ ⬘ 共兲y兴其, e⬜共y,t兲 = i1exp关− k⬜
再
冎
k⬘储 共兲 sin关t − k⬘储 共兲y兴 E 共兲. ⬙ 0
These relations involve the real parts of the appropriate complex refractive indices,
共2.30b兲
For the two modes of propagation we obtain from Eqs. 共2.3兲 and 共2.26兲 the following expressions for the electric field vectors:
⬙ 共兲y兴 h⬜共y,t兲 = − i3exp关− k⬜
k⬘储 共兲 cos关t − k⬘储 共兲y兴 0
These are equations of damped positive traveling waves, i.e., waves along the positive 2 共or y 兲 axis, with phase velocities,
= i1k储共兲E储共兲
H⬜共兲 = 兩H⬜共兲兩 =
+
再
⬘ 共兲 k⬜ ⬘ 共兲y兴 cos关t − k⬜ 0
冎
⬙ 共兲 k⬜ ⬘ 共兲y兴 E⬜共兲, sin关t − k⬜ 0
共2.32c兲
B. Propagation within the test cell
The analysis of the passage of radiation through the test cell is made difficult by the presence of the windows 共see Fig. 2兲 which give rise to multiple reflections within the windows themselves and within the sample. An exact analysis would require the calculation of the reflection and transmission coefficients at each interface in the cell. Such calculations would require detailed knowledge of the optical properties of the windows of the cell and its contents. Unfortunately this information is lacking. However, as we shall show, it is possible to evade this difficulty when the sample is strongly absorbing.
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174901-6
J. Chem. Phys. 122, 174901 共2005兲
B. K. P. Scaife and J. K. Vij
The electric field within the sample is the sum of the fields of a succession of damped forward and backward traveling waves. We shall take the origin of the coordinates to be just inside the sample as indicated in Fig. 2. The sample lies between the two 1-3 共x-z兲 planes passing through the points y = 0, and y = L, L being the width of the cell. We shall treat the interfaces as infinite planes. It is permissible to consider each mode of propagation separately. We shall, first, concentrate on the mode in which the electric field is perpendicular to the 3 共or z兲 axis. If the amplitude of the electric field entering, and just inside, the sample is denoted by E⬜共,0兲 = E⬜共兲 = i1E⬜共兲,
共2.40兲
then in view of the results obtained in Sec II A, the electric field on some plane y within the sample may be expressed in the following manner: e⬜共y,t兲 = i1Re E⬜共,y兲t ,
H⬜共,y兲 2 = − k⬜ 共兲E⬜共,y兲. y
In the following section, where we treat the Poynting vector, we shall require the value of
* * E⬜共,y兲H⬜ 共,y兲 = H⬜ 共,y兲 E⬜共,y兲 y y + E⬜共,y兲
= − 0兩H⬜共,y兲兩2 + =−
E⬜共,y兲 = E⬜共兲兵− k⬜y + R⬜共兲− k⬜共2L−y兲 共2.42兲
and R⬜共兲 takes account of the reflection at the samplewindow interface, including the multiple reflections within the window. In a similar way we have for the magnetizing force vector h⬜共y,t兲 = − i3Re H⬜共,y兲
t
共2.43兲
,
+ ¯ 其.
共2.44兲
Note that the odd powers of R⬜共兲 now have a negative sign because these terms arise from backward waves for which the magnetizing force vector is the reverse of that in a forward wave. As we shall later require the derivatives of E⬜共 , y兲 and of H⬜共 , y兲 with respect to y, it is convenient to give their values here
E⬜共,y兲 = E⬜共兲− k⬜yC⬜,E共,y兲
共2.50a兲
and k ⬜共 兲 E⬜共兲− k⬜yC⬜,H共,y兲. 0
共2.50b兲
The coefficients C⬜,E共 , y兲, and C⬜,H共 , y兲 have the values,
冋
C⬜,E共,y兲 = 1 + R⬜共兲
关R⬜共兲 + 2k⬜y兴−2k⬜L 兵1 − 关R⬜共兲兴2−k⬜L其
册
共2.51a兲 and
冋
C⬜,H共,y兲 = 1 + R⬜共兲
册
关R⬜共兲 − 2k⬜y兴−2k⬜L . 兵1 − 关R⬜共兲兴2−k⬜L其
It is clear from these last equations that the sum
− R⬜共兲− k⬜共2L−y兲 + 关R⬜共兲兴2− k1共2L+y兲 − 关R⬜共兲兴3− k⬜共3L−y兲 + 关R⬜共兲兴4− k⬜共3L+y兲 + ¯ 其
=
共2.51b兲
E⬜共,y兲 = †− k⬜共兲‡E⬜共兲兵− k⬜y y
共2.45兲
and using Eq. 共2.30a兲 we obtain
E⬜共,y兲 = − 0H⬜共,y兲. y
共2.49兲
H⬜共,y兲 = H⬜共兲− k⬜yC⬜,H共,y兲
+ 关R⬜共兲兴2− k⬜共2L+y兲 − 关R⬜共兲兴3− k⬜共3L−y兲 + 关R⬜共兲兴
⬘ 共兲k⬜ ⬙ 共兲 2k⬜ 1 兩E⬜共,y兲兩2 − †共0兲2兩H⬜共,y兲兩2 0 0
Note that the first term in the second part of this equation is entirely real, and the second term is entirely imaginary. The series which appear in Eqs. 共2.42兲 and 共2.44兲 can be summed, and we find it possible to write these equations in the following way:
k ⬜共 兲 E⬜共兲兵− k⬜y − R⬜共兲− k⬜共2L−y兲 0 4 − k⬜共3L+y兲
*2 k⬜ 共兲 兩E⬜共,y兲兩2 0
⬘ 共兲兴2 − 关k⬜ ⬙ 共兲兴2其兩E⬜共,y兲兩2‡. − 兵关k⬜
where, in view of Eq. 共2.30a兲, H⬜共,y兲 =
共2.48兲
* E⬜共,y兲H⬜ 共,y兲 y
in which
+ 关R⬜共兲兴4− k⬜共3L+y兲 + ¯ 其
* H 共,y兲 y ⬜
where * indicates the complex conjugate. By means of Eqs. 共2.46兲 and 共2.47兲 we get
共2.41兲
+ 关R⬜共兲兴2− k⬜共2L+y兲 + 关R⬜共兲兴3− k⬜共3L−y兲
共2.47兲
共2.46兲
In the same way we find from Eqs. 共2.42兲 and 共2.44兲 that
冋
C⬜,E共,y兲 + C⬜,H共,y兲 = 2 1 +
关R⬜共兲兴2−2k⬜L 兵1 − 关R⬜共兲兴2−k⬜L其
册
共2.52兲 and is independent of y. Finally we note that the relations for the mode in which the electric vector is parallel to the 3 共or z兲 axis may be obtained from the foregoing equations by replacing the subscript ⬜ by the subscript 储 throughout, and by changing the vector −i3 in Eq. 共2.43兲 to the vector i1.
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174901-7
J. Chem. Phys. 122, 174901 共2005兲
Infrared transmission spectroscopy of liquid crystals
⫻C⬜,H共,y兲−k⬜y + i1k储共兲E储共兲C储,H共,y兲−k储y其 = 共0兲−1Re tEin共兲 ⫻兵− i3sin k⬜共兲F⬜,E共兲C⬜,H共,y兲−k⬜y + i1cos k储共兲F储,E共兲C储,H共,y兲−k储y其.
共3.4b兲
The time average over one cycle, of duration T = 2 / , of the product of the two sinusoidal functions of time, a共t兲 = Re A共兲t
FIG. 5. The total electric vector E0 lies in the 1-3 plane and makes an angle with the 3 axis.
= 兩A共兲兩Re ␣共兲t = 兩A共兲兩cos关t + ␣共兲兴,
III. THE POYNTING VECTOR
The time-dependent Poynting vector s共r , t兲 is defined by the equation s共r,t兲 ⬅ e共r,t兲 ⫻ h共r,t兲
= 兩B共兲兩Re 共兲t
共3.1兲
and is a measure of the power-flux density in an electromagnetic field. Equation 共3.1兲 implies that for a plane electromagnetic wave, the Poynting vector s is parallel to the propagation vector k. To calculate the Poynting vector in the test cell we must take into consideration the angle between the electric vector of the incoming radiation and the 3 共or z兲 axis of the test cell. Thus the electric field, Ein共 , 兲 = E0共兲, in the wave approaching the test cell has two components 共see Fig. 5兲, Ein共, 兲 = 共i1sin + i3cos 兲E0共兲.
b共t兲 = Re B共兲t = 兩B共兲兩cos关t + 共兲兴, is given by a共t兲b共t兲T ⬅
E⬜共,0兲 = E⬜共兲 = i1F⬜,E共兲E0共兲sin ,
共3.3a兲
E储共,0兲 = E储共兲 = i3F储,E共兲E0共兲cos ,
共3.3b兲
in which the coefficients F⬜,E共兲 and F储,E共兲 take account of the properties of the windows and of the window-sample interface. The magnitudes of the magnetic vectors, H⬜共兲 and H储共兲, may be calculated from the corresponding electric vectors by means of Eqs. 共2.30兲, and their orientations are indicated in Fig. 4. Using the relations obtained in Sec. II B, we find that the total electric field vector within the sample is e共r,t兲 = Re兵i1E⬜共兲C⬜,E共,y兲关共t−k⬜y兲兴
T
dta共t兲b共t兲
0
= 21 Re A*共兲B共兲 = 21 兩A共兲兩兩B共兲兩cos关␣共兲 − 共兲兴.
共3.6兲
With this result we find that the time average of the Poynting vector with e and h given by Eqs. 共3.4a兲 and 共3.4b兲 is S共y兲 = i2S共y兲 = e共r,t兲 ⫻ h共r,t兲T = i2 2 „0…−1Re兩E0共兲兩2兵−2k⬜⬙ ysin2 1
* * ⫻关兩F⬜,E共兲兩2k⬜ 共兲C⬜,E共,y兲C⬜,H 共,y兲兴
+ −2k⬙储 ycos2关兩F储,E共兲兩2k*储 共兲C储,E共,y兲C*储,H共,y兲兴其. 共3.7兲 We may assume that E0共兲 is a real quantity so that S共y兲 =
1 2
冑
0 兩E 共兲兩2关cos2G储共,y兲 + sin2G⬜共,y兲兴 0 0 共3.8兲
G储共,y兲 = „c0/…exp关− 2k⬙储 共兲y兴
= Re tEin共兲兵i1sin F⬜,E共兲C⬜,E共,y兲−k⬜y
⫻兩F储,E共兲兩2Re关k*储 共兲C储,E共,y兲C*储,H共,y兲兴
共3.4a兲
= exp关− 2k⬙储 共兲y兴
and the magnetizing force vector is
* * ⫻兩F储,E共兲兩2Re k−1 0 关k 储 共兲C 储,E共,y兲C 储,H共,y兲兴
h共r,t兲 = Re兵− i3H⬜共兲C⬜,H共,y兲关共t−k⬜y兲兴
共3.9a兲
+ i1H储共兲C储,H共,y兲关共t−k储y兲兴其 = 共0兲−1Re t兵− i3k⬜共兲E⬜共兲
冕
with
+ i3E储共兲C储,E共,y兲关共t−k储y兲兴其 + i3cos F储,E共兲C储,E共,y兲−k储y其
1 T
= 21 Re A共兲B*共兲
共3.2兲
Each component of this approaching wave creates its own plane wave with propagation vectors: k⬜共兲 = i2k⬜共兲 and k储共兲 = i2k储共兲. The magnitudes of the electric vectors of the waves just entering the test cell will be directly proportional to the components of E0共兲. Therefore we write
共3.5兲
and
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174901-8
J. Chem. Phys. 122, 174901 共2005兲
B. K. P. Scaife and J. K. Vij
⬙ 共兲y兴 G⬜共,y兲 = „c0/…exp关− 2k⬜ * * ⫻兩F⬜,E共兲兩2Re关k⬜共 兲C⬜,E共,y兲C⬜,H共,y兲兴
⬙ 共兲y兴 = exp关− 2k⬜ * * ⫻兩F⬜,E共兲兩2Re k−1 0 关k⬜共兲C⬜,E共,y兲C⬜,H共,y兲兴.
共3.9b兲 The mean-power density in the beam approaching the test cell is Pin共, 兲 = Pref共, 兲 =
1 2
冑
0 兩E 共兲兩2 , 0 0
共3.10兲
so that the mean-power density entering the sample is S共0兲 = Pin共, 兲关cos2 G储共,0兲 + sin2 G⬜共,0兲兴 共3.11兲 and the mean-power density leaving the sample is S共L兲 = Pin共, 兲关cos2 G储共,L兲 + sin2 G⬜共,L兲兴. 共3.12兲 The mean-power density emerging from the cell, Pout共 , 兲, will be directly proportional to S共L兲 but slightly reduced in intensity by the effect of the window of the test cell. Hence we write
FIG. 6. The Poynting vector s parallel to the 2 axis, decreases in magnitude as the wave progresses from the 1-3 plane at y to the 1-3 plane at y + ␦y.
共3.7兲. The magnitude of S at the plane 共y + ␦y兲 may be expressed in terms of its magnitude at the plane y by means of the Taylor expansion, S共y + ␦y兲 = S共y兲 + ␦y
S共y兲 + 共…兲. y
The mean power absorbed in the slab of thickness ␦y and of unit 共1 m2兲 cross-sectional area is W共y兲␦y = S共y兲 − S共y + ␦y兲
Pout共, 兲 = Pin共, 兲†cos w储共兲G储共,L兲 2
+ sin w⬜共兲G⬜共,L兲‡. 2
Tcell共, 兲 =
⬵ S共y兲 − S共y兲 − ␦y
共3.13兲
The factors w储共兲 and w⬜共兲 take account of the small absorption of energy as the radiation passes through the exit window of the test cell. Recalling Eq. 共1.4兲, we see that Eq. 共3.13兲 leads to the following expression for the transmittance of the test cell: Pout共, 兲 = †⌳储共兲cos2 + ⌳⬜共兲sin2‡, Pin共, 兲 共3.14兲
共4.1兲
S共y兲 S共y兲 ⬵ − ␦y . y y
In the limit when ␦y is vanishingly small we obtain W共y兲 = −
S共y兲 . y
共4.3兲
Here W共y兲 is the mean power absorbed per unit volume on the plane y. It is convenient here to split the time-averaged Poynting vector into two parts, S储共y兲 and S⬜共y兲, for the two modes of propagation, so that in the notation of Sec. II B, we have * 共,y兲 S⬜共y兲 = e⬜共y,t兲h⬜共y,t兲T = 21 Re E⬜共,y兲H⬜
where ⌳储共兲 = exp†− 2k⬙储 共兲L‡w储共兲 ⫻兩F储,E共兲兩 Re 2
* * k−1 0 †k 储 共兲C 储,E共,L兲C 储,H共,L兲‡
共3.15a兲
共4.4兲
with a similar expression for S储共y兲. From Eq. 共2.49兲 and 共4.4兲 we get W⬜共y兲 = −
and
S⬜共y兲 y
= − 21 Re
⬙ 共兲L‡w⬜共兲 ⌳⬜共兲 = exp†− 2k⬜ * * ⫻兩F⬜,E共兲兩2Re k−1 0 †k⬜共兲C⬜,E共,L兲C⬜,H共,L兲‡.
=
共3.15b兲
The form of Eq. 共3.14兲 is that first proposed by Jang et al.6 Spatial dispersion is ignored in this analysis.14
共4.2兲
* †E⬜共,y兲H⬜ 共,y兲‡ y
⬘ 共兲k⬜ ⬙ 共兲 1 2k⬜ 兩E⬜共,y兲兩2 . 2 0
共4.5兲
By means of Eq. 共2.28b兲 we see that W⬜共y兲 = 21 01⬙共兲兩E⬜共,y兲兩2
共4.6兲
and hence IV. THE DENSITY OF POWER ABSORPTION
Consider two parallel planes perpendicular to the 2 共or y兲 axis and separated by a small distance ␦y, as shown in Fig. 6. The time-averaged Poynting vector S was defined in Eq.
W共y兲 = W储共y兲 + W⬜共y兲 = 21 0†3⬙共兲兩E储共,y兲兩2 + 1⬙共兲兩E⬜共,y兲兩2‡
共4.7兲
16
in agreement with standard dielectric theory.
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174901-9
J. Chem. Phys. 122, 174901 共2005兲
Infrared transmission spectroscopy of liquid crystals
␣s = ␣⬘共0兲
共5.4兲
and the angular brackets indicate an average given by 具共…兲典 =
冕 冕 2
d␥
d sin 共, ␥兲共…兲.
共5.5兲
0
0
Similarly, for e = i2e2 we have p1,2 = 1,20e2 = FIG. 7. The angular coordinates  and ␥ specify the orientation of a molecule with respect to the axes 1, 2, and 3.
共5.6a兲
= 共1,2 − 1兲0e2 , p2,2 = 2,20e2 =
V. MICROSCOPIC CONSIDERATIONS
N 具␣se2sin ␥ cos ␥ sin2典 V
N 具␣se2sin2␥ sin2典 = 共2 − 1兲0e2 , V
As a model we assume that the medium consists of long, rodlike molecules with only an axial electrical polarizability,
␣共兲 = ␣⬘共兲 − ı␣⬙共兲.
共5.1兲
The problem of the local field arises;16 given the postulated shape of the molecules, it is not unreasonable to assume that the local field is equal to the macroscopic electric field. 共The depolarizing factor for an exceedingly prolate spheroid is negligibly small.16兲 If the angular coordinates of a molecule are  and ␥ as in Fig. 7, the orientational distribution function 共 , ␥兲 will be taken to be7
共5.6b兲 p3,2 = 3,20e2 =
+ E sin 2 sin ␥ + F sin2 sin 2␥兴,
and for e = i3e3, p1,3 = 1,30e2 =
p2,1 = 2,10e1 =
共5.3a兲
N 具␣se1cos ␥ sin ␥ sin 典 = 共3,1 − 1兲0e1 . V
N 具␣se3sin ␥ sin  cos 典 V 共5.7b兲
= 共2,3 − 1兲0e3 , p3,3 = 3,30e2 =
N 具␣se3cos2典 = 共3 − 1兲0e3 . V
共5.7c兲
Clearly
␣, = ,␣,
␣, = ,␣ .
共5.8兲
For an electric field 共5.9兲
共5.10兲
p = i1 p1 + i2 p2 + i3 p3 ,
where the components of p are given by the matrix equation:
冤冥冤 p1
共5.3b兲
p3,1 = 3,10e1 =
p2,3 = 2,30e2 =
the induced-polarization vector
N 具␣se1cos ␥ sin ␥ sin2典 V
= 共2,1 − 1兲0e1 ,
共5.7a兲
e = i 1e 1 + i 2e 2 + i 3e 3 ,
N p1,1 = 1,10e1 = 具␣se1 cos2␥ sin2典 V = 共1 − 1兲0e1 ,
N 具␣se3cos ␥ sin  cos典 V
= 共1,3 − 1兲0e3 ,
共5.2兲
Here A, B, C, D, E, and F are the expansion coefficients and serve as the order parameters with the maximum value of 1 in completely ordered systems. A measures the polar order and for the ferroelectric phase ⫽0 in the unwound state. B measures the order relative to the axis 3. C, D, E, and F are the four order parameters measuring the quadrupolar order. P2共cos 兲 = 共1 / 2兲共3 cos2 − 1兲 is a Legendre polynomial. For a static electric field e = i1e1, the polarization components are
共5.6c兲
= 共3,2 − 1兲0e2 ,
4共, ␥兲 = 1 + 3A sin  sin ␥ + 5BP2共cos 兲 + 共15/4兲关C sin 2 cos ␥ + D sin2 cos 2␥
N 具␣se2sin ␥ sin  cos 典 V
共5.3c兲
In these equations N denotes the number of molecules in volume V,
共p1,1 + p1,2 + p1,3兲
p2 = 共p2,1 + p2,2 + p2,3兲 p3 共p3,1 + p3,2 + p3,3兲
冤
1,1 = 0 2,1 3,1
1,2 2,2 3,2
冥 1,3 2,3 3,3
冥冤 冥 e1
e2 . e3
共5.11兲
On evaluating the averages in Eqs. 共5.3兲, 共5.6兲, and 共5.7兲 by means of Eq. 共5.5兲 共see the Appendix兲 we find that
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174901-10
冤
1,1 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 N␣s = 0V
J. Chem. Phys. 122, 174901 共2005兲
B. K. P. Scaife and J. K. Vij
冤
†
1−B 3
冥 +
A共, /2兲 = A⬜共兲 = − log10n1⬘共兲 + 2共2L/0兲n1⬙共兲log10. D 2
F 2 C 2
‡
†
F 2 1−B 3
−
D 2
E 2
‡
†
C 2 E 2 1+2B 3
‡
冥
共6.2b兲 .
共5.12兲
The condition that the susceptibility matrix, [␣,], be diagonal is that 共5.13兲
C=E=F=0
冋 册 冋 册 冋 册
N␣共兲 1 − B D + , 0V 3 2
N␣共兲 1 − B D 2共 兲 − 1 = − , 0V 3 2 3共 兲 − 1 =
N␣共兲 1 + 2B . 0V 3
共5.14a兲
共5.14b兲
兩E⬜共,y兲兩2 = 兩E⬜共兲兩210−A⬜共兲共y/L兲 .
共6.3兲
兩E储共,y兲兩2 = 兩E储共兲兩2exp关− 2k⬙储 共兲y兴
⬙ 共兲y兴. 兩E⬜共,y兲兩2 = 兩E⬜共兲兩2exp关− 2k⬜
共6.4兲
For Eqs. 共6.3兲 and 共6.4兲 to be compatible it is necessary that 共5.14c兲
Hence the susceptibility of the sample in its isotropic state is given by
iso共兲 ⬅ iso共兲 − 1 = 31 †1共兲 + 2共兲 + 3共兲‡
冋
兩E储共,y兲兩2 = 兩E储共兲兩210−A储共兲共y/L兲 ,
With the approximations introduced in this section, our versions of these equations, based on Eq. 共2.50a兲, read
and under this condition we get 1共 兲 − 1 =
Kocot et al.7 assume that the modulus squared of the electric field within the sample varies with distance according to the equations:
=
N␣共兲 1 − B D 1 − B D 1 + 2B + + − + 30V 3 2 3 2 3
=
N␣共兲 30V
册
A储共兲 = 2k⬙储 共兲L log10,
共6.5兲 This condition could be satisfied if the absorption was so strong as to make the log10n⬘共兲 term, in Eq. 共6.2兲, small in comparison with the term in n⬙共兲. Kocot et al.7 also derived an expression for Pabs共 , 兲, the mean-power density absorbed by the cell, which seems to be in error. From the discussion in Sec. I and the definition in Eq. 共1.4兲 it is clear that with Pabs共, 兲 Pin共, 兲 − Pout共, 兲 = = 1 − Tcell共, 兲, 共6.6兲 Pin共, 兲 Pin共, 兲
共5.15兲
as expected. Explicit expressions for k⬜共兲 and k储共兲 in terms of iso共兲, B, and D may be obtained by combining Eq. 共2.17兲 with the results of this section.
⬙ 共兲L log10. A⬜共兲 = 2k⬜
then 1 − Tcell共, 兲 = 1 − 关⌳储共兲cos2 + ⌳⬜共兲sin2兴 = 关1 − ⌳储共兲兴cos2 + 关1 − ⌳⬜共兲兴sin2 , 共6.7兲
VI. DISCUSSION
In order to simplify the discussion we shall assume that the windows of the test cell are only weakly reflecting and hence that their transmittances are close to unity. Under these assumptions Eqs. 共3.15a兲 and 共3.15b兲 simplify considerably and take the approximate forms, ⌳储共兲 ⬵ k−1 储 共 兲exp关− 2k ⬙ 储 共 兲L兴 0 k⬘ ⬵ n3⬘共兲exp关− 2共2/0兲n⬙3共兲L兴,
共6.1a兲
which differs from the result obtained by Kocot et al. 关Ref. 7, Eq. 共8兲兴. For the case of weak absorption, we have the approximation, Pabs共, 兲 ⬵ Pin共, 兲
2L 兵关2n⬘3共兲n3⬙共兲兴cos2 0
+ 关2n⬘共兲n1⬙共兲兴sin2其 = Pin共, 兲
2L 关⬙共兲cos2 + ⬙1共兲sin2兴. 0 3 共6.8兲
⬘ 共兲exp关− 2k⬜ ⬙ 共兲L兴 ⌳⬜共兲 ⬵ k−1 0 k⬜ ⬵ n⬘1共兲exp关− 2共2/0兲n1⬙共兲L兴.
共6.1b兲
The absorbance A共 , 兲, defined by Eq. 共1.5兲, has the following particular values: A共,0兲 = A储共兲 = − log10n3⬘共兲 + 2共2L/0兲n⬙3共兲log10, 共6.2a兲
When the sample is isotropic this equation becomes 共iso兲 共, 兲 = Pin共, 兲 Pabs
2L ⬙ 共兲. 0 iso
共6.9兲
On dividing Eq. 共6.8兲 by Eq. 共6.9兲, we find, by using Eqs. 共5.14兲 and 共5.15兲, that
Downloaded 11 May 2009 to 134.226.1.229. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174901-11
J. Chem. Phys. 122, 174901 共2005兲
Infrared transmission spectroscopy of liquid crystals
Pabs共, 兲 共iso兲 Pabs 共, 兲
再
冋
= 关1 + 2B兴cos2 + 1 − B + = 1 + B共2 − 3 sin2兲 +
册 冎
3D sin2 2
3D 2 sin . 2
ACKNOWLEDGMENTS
One of the authors 共JKV兲 thanks the Science Foundation of Ireland for a partial funding of this reseach under Grant 共02/IN.1/I031兲, S. Sanvito for discussions, and R. Korlacki for converting the diagrams from McIntosh to PC. APPENDIX: RESULTS OF MATHEMATICAL AVERAGES OF TRIGNO-METRIC FUNCTIONS OF THE ANGULAR COORDINATES OF A MOLECLUE
We list here the values of the averages used in the calculation of ␣, in Sec. V:
冋
册
具sin2 sin2␥典 =
1 3D 1−B− , 3 2
具cos2典 = 31 共1 + 2B兲,
VII. CONCLUSIONS
We have developed the theory of an electromagnetic homogeneous plane-wave in a semi-infinite, optically anisotropic and absorbing medium. Based on this, and from first principles, we have derived the empirical equation first proposed by Jang, Park, Kim, and Clark6 for the IR absorbance as a function of the angle of polarization for normal incidence of an IR beam in a ferroelectric liquid crystalline sample aligned in the homogeneous configuration and the sample confined to lie between the windows of a cell. We find that this equation is not subject to the limitation that the IR birefringence of the sample should be zero. In the derivation, we considered the elements of the dielectric tensor to be complex and frequency dependent but neglected the explicit dependence on the wave vector. We thus assume that all values of the wavevector are allowed for the various frequencies. This is justified by the fact the smectic layer thickness 共⬃3 nm兲 is much smaller that the shortest wavelength in the IR beam 共⬃3 m兲. The absorbances along the direction of polarization of the IR beam, and normal to it, are expressed in terms of the properties of the materials and the dimensions of the cell. We remark that an earlier simplified attempt by Kocot, Wrzalik, and Vij7 to derive this formula was based on certain assumptions and in some cases the equations were in error. It has been shown that the order parameters in certain cases can also be calculated from the three different orthogonal components of the relative permittivity.
册
1 3D 1−B+ , 3 2
共6.10兲
This equation agrees with that obtained by Kocot et al. for D = 0 关Ref. 7, Eq. 共12兲兴.
冋
具sin2 cos2␥典 =
共A1兲
共A2兲
共A3兲
具sin2 sin ␥ cos ␥典 =
F , 2
共A4兲
具sin  cos  sin ␥典 =
E , 2
共A5兲
具sin  cos  cos ␥典 =
C . 2
共A6兲
1
B. J. Bulkin, Liquid Crystals, the Fourth State of Matter, edited by F. D. Saeva 共Dekker, New York, 1975兲. 2 A. Kocot, G. Kruk, R. Wrzalik, and J. K. Vij, Liq. Cryst., 12, 1005 共1992兲. 3 K. Merkel, A. Kocot, J. K. Vij, G. H. Mehl, and T. Meyer, J. Chem. Phys. 121, 5012 共2004兲. 4 E. Hild, A. Kocot, J. K. Vij, and R. Zentel, Liq. Cryst. 16, 783 共1994兲. 5 K. H. Kim, K. Ishikawa, H. Takezoe, and A. Fukuda, Phys. Rev. E 51, 2166 共1995兲. 6 W. G. Jang, C. S. Park, J. E. Maclennan, K. H. Kim, and N. A. Clark, Ferroelectrics 180, 213 共1996兲. 7 A. Kocot, R. Wrzalik, and J. K. Vij, Liq. Cryst. 21, 147 共1996兲. 8 A. Kocot, R. Wrzalik, B. Orgasinska, T. S. Perova, J. K. Vij, and H. T. Nguyen, Phys. Rev. E 59, 551 共1999兲. 9 A. A. Sigarev, J. K. Vij, Yu. P. Panarin, and J. W. Goodby, Phys. Rev. E 62, 2269 共2000兲; A. A. Sigarev, J. K. Vij, R. A. Lewis, M. Hird, and J. W. Goodby, ibid. 68, 031707 共2003兲. 10 W. G. Jang, C. S. Park, K. H. Kim, M. A. Glaser, and N. A. Clark, Phys. Rev. E 62, 5027 共2000兲. 11 W. G. Jang, C. S. Park, and N. A. Clark, Phys. Rev. E 62, 5154 共2000兲. 12 A. Kocot, J. K. Vij, and T. S. Perova, Adv. Chem. Phys. 113, 203 共2000兲; T. S. Perova, J. K. Vij, and A. Kocot Adv. Liq. Cryst. in a special issue of Advances in Chemical Physics, edited by J. K. Vij 共series editors I. Prigogine and S. Rice兲 共John Wiley & Sons, New York兲 113, 341 共2000兲. 13 L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. 共Pergamon, Oxford 1984兲, Chap. XI. 14 H. Yuan, W. E. Palffy-Muhoray, and P. Palffy-Muhoray, Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 358, 311 共2001兲. 15 H. Yuan, W. E. Palffy-Muhoray, and P. Palffy-Muhoray, Phys. Rev. E 61, 3264 共2000兲. 16 B. K. P. Scaife, Principles of Dielectrics, Revised ed. 共Clarendon, Oxford, 1998兲.
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