Berreman thickness

Once the reflectance at the band maximum is reduced to zero (/ 0), the band depth drops and the band shape becomes distorted, narrowing if the spectrum is represented in absorbance [- log(/ // o)] units and broadening in AR (or AR/R) units. To designate the onset of significant distortions in j9-polarized spectra, the term Berreman thickness ds is sometimes applied [4]. At the air-metal interface, the Berreman thickness can be approximated by [4]... 

Another important observation from Fig. 3.14 is that at small thicknesses the intensity of the vlo band depends linearly on the film thickness, in agreement with the thin-fllm approximation [Eq. (1.82)]. This linear region extends to approximately half the Berreman thickness and depends upon the angle of incidence in the same way as dg. The linear range of the band intensity in absorbance is somewhat broader than that in reflectance, as illustrated by comparing curves 1 and 4. This implies that absorbance units should be used in any quantitative analysis of thin films, rather than reflectivity units. 

The Berreman thickness drops abruptly if either the angle of incidence or the refractive index of the immersion medium is increased. If for a grazing angle of incidence, d > dB,iht angle should be decreased so that d = ds. 

The spectra from strong oscillators have special features which are different from those from metallic and dielectric substrates. Different structures in tanf and A are observed on a metallic substrate, dependent on the thickness of the film (Fig. 4.65). For very thin films up to approximately 100 nm the Berreman effect is found near the position of n = k and n < 1 with a shift to higher wavenumbers in relation to the oscillator frequency. This effect decreases with increasing thickness (d > approx. 100 nm) and is replaced by excitation of a surface wave at the boundary of the dielectric film and metal. The oscillator frequency (TO mode) can now also be observed. On metallic substrates for thin films (d < approx. 2 pm) only the 2-component of the electric field is relevant. With thin films on a dielectric substrate the oscillator frequency and the Berreman effect are always observed simultaneously, because in these circumstances all three components of the electric field are possible (Fig. 4.66). 

Fig. 4.65. Different spectral features of tanf for a strong model oscillator at 1000 cm" on a metal substrate. The TO mode (1000 cm" ), Berreman effect (1050 cm" ), and excitation ofa surface wave (1090 cm" ) are seen for different 1150 thicknesses - 1, 5, 10, 50,100, 500, and 1000 nm.

It is usually possible to investigate very thin films (up to the subnanometer range) by use of infrared wavelengths, which are much greater than the thickness of the film (a factor of 10000) because of the interference optics of the strong oscillator (Berreman effect). 

Figure 6.4-7 Berreman effect observed with a ca. 22 nm thick layer of SiOi on Si substrate.

An outstanding gain in sensitivity, allowing one to identify and quantify surface layers of some nanometers thickness, can be observed for strong oscillators due to the Berreman effect. 

Fig. 4.1.15. First and second order reflexion spectra of a cholesteric liquid crystal film (0.45 0.55 mole fraction mixture of 4 -bis(2-methylbutoxy)-azoxybenzene and 4,4 -di-n-hexoxyazoxybenzene) 15 pitch lengths or 11.47 on thick. Angle of incidence 45°. Polarizer and analyser are parallel to the plane of reflexion for and normal to it for measurements. The small oscillations are interference fringes from the two cholesteric-glass interfaces. (After Berreman and Scheffer. )...

They were computed for a single Lorentzian absorption band, with a peak molar extinction coefficient of 17.000 liter mole cm , and for a liquid crystal thickness of 1 nm an order parameter of 0.7 has been assumed for the liquid crystal. The curves were calculated using a computer program written by D. W. Berreman. ... 

A definition of these angles is given in Fig, 1, The deformation profile is dependent on the dielectric constants C and Ej., the elastic constants for splay, twist and bend Kn, K22> 33 the total twist (90-23q), the tilt angle at the surface of the substrates ao, and the applied voltage. The optical response depends in addition on the refractive indices ne and viq and the ratio of wavelength to cell thickness. For display applications a finite tilt at the surfaces is required to avoid areas of opposite tilt. Therefore the deformation profiles are calculated for various combinations of K33/K11, Ae/ej and using Berreman s program. All calculations are performed for 10 im cells and a pretilt ao=l°. 

We divide the cholesteric film into N slabs with thicknesses Az. The Berreman vectors at the boundaries between the slabs are ... 

Cell thickness-dependence of the reflection of a cholesteric liquid crystal in the planar state. The pitch of the liquid crystal is P = 350 nm. The refractive indices of the liquid crystal are tig = 1 -7 and = 1.5. The liquid crystal is sandwiched between two glass plates with the refractive index = 1.6. The incident light is circularly polarized with the same helical handedness as the liquid crystal. Neglect the reflection from the glass-air interface. Use two methods to calculate the reflection spectrum of the liquid crystal with the following cell thicknesses P, 2P, 5P and lOP. The first method is the Berreman 4x4 method and the second method is using Equation (2.186). Compare the results from the two methods. 

Berreman and Heffner [59] considered the cholesteric Grandjean texture with tilted director orientation on the boundaries, Fig. 6.16. In the absence of the tilt, the free energy g is minimum at the following thicknesses d of the Cano wedge ... 

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