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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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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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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 = − ␻2␮0D0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 = − ␻2␮0D0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␧0␮0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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J. Chem. Phys. 122, 174901 共2005兲


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

k20␧1 − 共k22 + k23兲

k 1k 2 k20␧2

k 1k 2 k 1k 3

−

共k21

k 1k 3 k23兲

+

k 2k 3 k20␧3 − 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

k20␧1 − k22 − k23

k 1k 2 k20␧2

k 1k 2 k 1k 3

−

k21

k 1k 3 −

k23

k 2k 3

k 2k 3 k20␧3 − 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

= k20␧3共␻兲 =

where we have used the facts that

2

␧ 1共 ␻ 兲 . ␧ 3共 ␻ 兲

= k20␧1共␻兲 =

2 −2 ␧1␧2␧3 − ␧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 − 4␧1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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J. Chem. Phys. 122, 174901 共2005兲


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 = − i3␻␮0H⬜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 = i1␻␮0H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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J. Chem. Phys. 122, 174901 共2005兲


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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J. Chem. Phys. 122, 174901 共2005兲


⫻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cos2␪G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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J. Chem. Phys. 122, 174901 共2005兲


⬙ 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 ␻␧0␧1⬙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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J. Chem. Phys. 122, 174901 共2005兲


␣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 = ␹1,2␧0e2 = 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 = ␹2,2␧0e2 =

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 = ␹3,2␧0e2 =

+ E sin 2␤ sin ␥ + F sin2␤ sin 2␥兴,

and for e = i3e3, p1,3 = ␹1,3␧0e2 =

p2,1 = ␹2,1␧0e1 =

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 = ␹3,3␧0e2 =

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 = ␹3,1␧0e1 =

p2,3 = ␹2,3␧0e2 =

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 = ␹1,1␧0e1 = 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


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


冤

†

1−B 3

冥 +

A共␻, ␲/2兲 = A⬜共␻兲 = − log10n1⬘共␻兲 + 2共2␲L/␭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 + + − + 3␧0V 3 2 3 2 3

=

N␣共␻兲 3␧0V

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

2␲L 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

2␲L 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2␲L/␭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

2␲L ␧⬙ 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


174901-11

J. Chem. Phys. 122, 174901 共2005兲


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.
