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 = hn
The description of most atomic processes such as absorption, fluorescence, and the photoelectric effect require the photon approach.
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[(2p/l)(x+vt)]
The velocity (v) is related to the frequency (n) by the equation:
v = ln
The electro-magnetic wave intensity (I) is proportional to the square of the amplitude:
I = K(ao)2
K is a constant of proportionality and depends upon the properties of the medium containing
the wave.
Polarisation of Light
linearly polarized
circularly 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
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 = sinq1 / sinq2 (angles of incident & refraction,
respectively)
sinq1 / sinq2 = n2 / n1
The velocity of a light wave changes when light enters a transparent
medium of different refractive index but not the frequency:
velocity = nl;
n = c / v = lvac / lsubs
Graphical change of wavelength
with change of n.
Total Internal Reflection
At the critical angle qc, 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 qc = n(low) / n(high)
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(q1 – q3) / sin (q1 + q3)]2
Rp = [tan(q1 – q3) / tan (q1 + q3)]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 Dn = ne-no
Double Refraction and Birefringence of an
Anisotropic Transparent Medium
q
The relationship between the magnitude
of n’e and the angle q that the ray makes
with the optic axis is:
1 / (n’e)2 = cos2q / no2 + sin2q / ne2
Snell’s Law: sinq1 / sinq2 = n2 / n1
Birefringence and the Indicatrix
Molecular Theory of Refractive Indices
Lorentz local field for an isotropic medium: Eloc= [(e + 2) / 3] E
e is the mean permittivity
Using e = 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 a / 3e0
where N is the number density (d NA / M ), a the
mean polarizability, and e0 = 8.86 10-12 As/Vm
Anisotropic Molecular Polarizability
ne2 – 1 / n2 + 2 = N a / 3e0
with n2 = 1/3 (ne2 + 2no2)
no2 – 1 / n2 + 2 = N a / 3e0
Defect Textures in Thermotropic Liquid Crystals
Schlieren Texture of a Nematic Phase
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
Deformations in Thermotropic Liquid Crystals
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.
Natural textures exhibited by calamitic LCs
(as seen between crossed polarizers)
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 Dphase
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
Paramorphotic textures associated with 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