Two models for the description of light The corpuscular theory of light stating that light can be regarded as a stream of particles of discrete energy called photons. Their energy E is defined by: E = h The wave theory of light stating that light can be treated as a wave with an electrical and magnetic field, each described by a vector. The magnitude of the electric field vector Y at position x and time t and the amplitude a o (constant) is given by: Y = a o sin[(2  )(x+vt)] The velocity (v) is related to the frequency ( ) by the equation: v = The description of most atomic processes such as absorption, fluorescence, and the photo- electric effect require the photon approach. The electro-magnetic wave intensity (I) is proportional to the square of the amplitude: I =  a    K is a constant of proportionality and depends upon the properties of the medium containing the wave.

Polarisation of Light circularly polarized linearly polarized

Four principal interactions of light with matter Ignoring fluorescence, the interactions of light with matter can be expressed thus: I o = I reflected + I scattered + I absorbed + I transmitted transparent material translucent material

Refractive Index and Polarizability of Matter

Refraction and the refractive index When light enters a transparent medium of different refractive index, n, it is refracted (Snell’s Law): n = sin    sin    angles of incident & refraction, respectively) sin  1 / sin  2 = n 2 / n 1 The velocity of a light wave changes when light enters a transparent medium of different refractive index but not the frequency: velocity =  n = c / v = vac  subs Graphical change of wavelength with change of n.

At the critical angle  c, the emerging ray travels exactly along the surface. Exceeding this angle results in total reflection (no light is lost). The critical angle is given by: sin  c = n(low) / n(high) Total Internal Reflection

Dispersion and Colour The refractive index of a transparent solid varies with wavelength. This is called dispersion.

Polarisation by Reflection The reflectivity or reflectance of a surface is given by: R s = [sin(  1 –  3 ) / sin (  1 +  3 )] 2 R p = [tan(  1 –  3 ) / tan (  1 +  3 )] 2 The Brewster’s angle

Birefringence of Optically Anisotropic Matter Birefringence A nematic phase, for example, is essentially a one-dimensionally ordered elastic fluid in which the molecules are orientationally ordered along the director. The nematic phase is birefringent due to the anisotropic nature of its physical properties. Thus, a light beam entering into a bulk nematic phase will be split into two rays, an ordinary ray and an extraordinary ray (along the director). These two rays will be deflected at different angles and travel at different velocities through the mesophase, depending on the principal refractive indices. If the extraordinary ray travels at a slower velocity than the ordinary ray, the phase has a positive birefringence. We can write for most optically uniaxial calamitic mesophases: n e > n o with  n = n e -n o

Double Refraction and Birefringence of an Anisotropic Transparent Medium The relationship between the magnitude of n’ e and the angle  that the ray makes with the optic axis is: 1 / (n’ e ) 2 = cos 2  / n o 2 + sin 2  / n e 2 Snell’s Law: sin  1 / sin  2 = n 2 / n 1 

Birefringence and the Indicatrix

Molecular Theory of Refractive Indices Lorentz local field for an isotropic medium: E loc = [(  + 2) / 3] E   is the mean permittivity Using  = n 2 derived from the Maxwell’s Equations, the Lorenz-Lorentz expression relates the refractive index to the mean molecular polarizability: n 2 – 1 / n = N  / 3  0 where N is the number density (d N A / M ),  the mean polarizability, and  0 = As/Vm n e 2 – 1 / n = N  / 3  0 n o 2 – 1 / n = N  / 3  0 with n 2 = 1/3 (n e 2 + 2n o 2 ) Anisotropic Molecular Polarizability

Schlieren Texture of a Nematic Phase Defect Textures in Thermotropic Liquid Crystals

Textures of a SmA Phase

Textures of a SmC Phase broken focal-conic schlieren

Textures of a Col h Phase

Mosaic Texture of a SmB Phase

Since the nematic phase can be treated as an elastic continuum fluid, three possible elastic deformations of its structure are possible: The splay deformation, the twist deformation, and the bend deformation. The elastic constants associated with them are k 11, k 22, and k 33, respectively. Deformations in Thermotropic Liquid Crystals

MesophaseHomogeneous (planar) alignment Homeotropic (orthogonal) alignment Mechanical shearingother Nematic Nschlierenextinct blackshears easilyBrownian flashes SmAfocal-conic, polygonal defects extinct blackshears to homeotropic Cubic D- phase extinct black viscousgrows in squares or rectangles SmCfocal-conic brokenschlieren (4 brushes)shears to schlierenBrownian motion SmBfocal-conicextinct blackshears to homeotropic Mosaic possible SmIfocal-conic brokenschlierenshears viscousschlieren diffuse Crystal Bmosaicextinct blackshears viscousgrain boundaries SmFmosaicschlieren, mosaicshears viscousgrain boundaries Crystal Jmosaic very viscousgrain boundaries Crystal Gmosaic very viscousgrain boundaries Crystal Emosaicshadowy mosaicvery viscousgrain boundaries Crystal Hmosaic very viscousgrain boundaries Crystal Kmosaic very viscousgrain boundaries Natural textures exhibited by calamitic LCs (as seen between crossed polarizers)

MesophaseParamorphotic textures SmCbroken focal-conic from SmA focal-conic; schlieren from SmA homeotropic; sanded schlieren from cubic D-phases SmIfocal-conic broken, chunky defects from SmA or C focal-conic; schlieren from schlieren SmC or SmA homeotropic SmBfocal-conic from SmA focal-conic; clear focal-conic defects from broken SmC focal-conic; extinct homeotropic from SmA homeotropic or SmC schlieren Crystal Bclear focal-conic from focal-conic SmA, B or C; homeotropic from homeotropic SmA and B or SmC schlieren SmFbroken focal-conic from focal-conic SmA, B, C, I or crystal B; schlieren mosaic from homeotropic SmA and B or SmC and I schlieren Crystal JBroken pseudo focal-conic fans, chunky from focal conic domains SmA, B, C, I, and F or crystal B; mosaic from homeotropic SmA, B, and crystal B or SmC and I schlieren or SmF schlieren mosaic Crystal G Crystal E Crystal H Crystal K Paramorphotic textures associated with calamitic LCs (as seen between crossed polarizers)