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 ao (constant) is given by:

Y = aosin[(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:Io = Ireflected + Iscattered + Iabsorbed + Itransmitted

transparentmaterial

translucentmaterial

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 sinangles of incident & refraction,

respectively)

sin1 / sin2 = n2 / n1

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 = vacsubs

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:

Rs = [sin(1 – 3) / sin (1 + 3)]2

Rp = [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:

ne > no with n = ne-no

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 = cos2 / no2 + sin2 / ne

2

Snell’s Law: sin1 / sin2 = n2 / n1

Birefringence and the Indicatrix

Molecular Theory of Refractive Indices

Lorentz local field for an isotropic medium: Eloc= [(+ 2) / 3] E

is the mean permittivity

Using = n2 derived from the Maxwell’s

Equations, the Lorenz-Lorentz expression relates

the refractive index to the mean molecular

polarizability: n2 – 1 / n2 + 2 = N / 30

where N is the number density (d NA / M ), the

mean polarizability, and 0 = 8.86 10-12 As/Vm

ne2 – 1 / n2 + 2 = N / 30 no

2 – 1 / n2 + 2 = N / 30

with n2 = 1/3 (ne2 + 2no

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 Colh 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 k11, k22, and k33, respectively.

Deformations in Thermotropic Liquid Crystals

Mesophase Homogeneous (planar) alignment

Homeotropic (orthogonal) alignment

Mechanical shearing other

Nematic N schlieren extinct black shears easily Brownian flashes

SmA focal-conic, polygonal defects

extinct black shears to homeotropic

Cubic D-phase

extinct black extinct black viscous grows in squares or rectangles

SmC focal-conic broken schlieren (4 brushes) shears to schlieren Brownian motion

SmB focal-conic extinct black shears to homeotropic

Mosaic possible

SmI focal-conic broken schlieren shears viscous schlieren diffuse

Crystal B mosaic extinct black shears viscous grain boundaries

SmF mosaic schlieren, mosaic shears viscous grain boundaries

Crystal J mosaic mosaic very viscous grain boundaries

Crystal G mosaic mosaic very viscous grain boundaries

Crystal E mosaic shadowy mosaic very viscous grain boundaries

Crystal H mosaic mosaic very viscous grain boundaries

Crystal K mosaic mosaic very viscous grain boundaries

Natural textures exhibited by calamitic LCs(as seen between crossed polarizers)

Mesophase Paramorphotic textures

SmC broken focal-conic from SmA focal-conic; schlieren from SmA homeotropic; sanded schlieren from cubic D-phases

SmI focal-conic broken, chunky defects from SmA or C focal-conic; schlieren from schlieren SmC or SmA homeotropic

SmB focal-conic from SmA focal-conic; clear focal-conic defects from broken SmC focal-conic; extinct homeotropic from SmA homeotropic or SmC schlieren

Crystal B clear focal-conic from focal-conic SmA, B or C; homeotropic from homeotropic SmA and B or SmC schlieren

SmF broken 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 J Broken 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)