Double Refraction Theory

Theory of Double Refraction

• When unpolarized light passes through a doubly refracting crystal like calcite or quartz, it splits into two refracted rays.
• One ray obeys Snell's law and is called the ordinary ray (O-ray).
• The other ray does not obey Snell's law and is not in the plane of incidence; it is called the extraordinary ray (E-ray).
• The O-ray is plane polarized in the principal plane, with vibrations perpendicular to the principal plane, represented by dots.
• The E-ray is also plane polarized in a plane perpendicular to the principal plane, with vibrations in the principal plane, represented by lines with arrow heads.
• O-rays travel with the same speed in all directions within the crystal.
• The refractive index for the O-ray has a single value.
• E-rays travel with a speed that varies with direction in the crystal.
• The refractive index for E-rays varies with direction.
• Birefringence is the difference between the refractive indices for O and E-rays (μo - μe).
• Along the optic axis, both O-rays and E-rays travel with the same speed.
• In uniaxial crystals, the wavefront of O-rays is spherical, and the wavefront of E-rays is ellipsoidal in nature.

Malus' Law

• Malus' law describes how the intensity of light transmitted through an analyzer changes with the angle between the transmission planes of the analyzer and polarizer.
• The intensity of emergent light varies as the square of the cosine of the angle (θ) between the transmission planes of the analyzer and polarizer: I = I₀cos²θ.
• I₀ represents the intensity of the incident plane-polarized light.
• When θ = 0° or 180°, cos²θ = 1, resulting in maximum intensity (I = I₀).
• When θ = 90°, cos²θ = 0, resulting in minimum intensity (I = 0).
• Unpolarized light intensity is reduced by 50% after passing through a polarizer.
• Ideal polarizers have a maximum transmission of 50%.

Retardation Plates

• Retardation plates are crystal plates of doubly refracting material that retard the motion of one of the refracted beams (O-ray or E-ray).
• Two types of retardation plates: quarter-wave plates and half-wave plates.

Quarter-Wave Plate

• A quarter-wave plate is a doubly refracting crystal with its optic axis parallel to the refracting faces.
• The thickness of the plate creates a path difference of λ/4 (or a phase difference of π/2) between the O-ray and E-ray.
• This occurs when monochromatic plane-polarized light of wavelength λ is incident normally on the surface.
• For calcite (a negative crystal) where the O-ray's velocity exceeds the E-ray's, the thickness is calculated to achieve the λ/4 path difference.
• t = λ / [4(μo - μe)]
• For quartz (a positive crystal), the thickness is calculated as t = λ / [4(μe - μo)].

Half-Wave Plate

• A half-wave plate is a doubly refracting crystal with its optic axis parallel to the refracting faces.
• The thickness creates a path difference of λ/2 (or a phase difference of π) between the O-ray and E-ray when monochromatic plane-polarized light of wavelength λ falls normally.
• In calcite crystals, the thickness is determined by t = λ / [2(μo - μe)].
• In quartz crystals, the thickness is determined by t = λ / [2(μe - μo)].

Optical Activity

• Optical activity is the phenomenon where a plane of polarized light rotates when passing through certain substances.
• These substances are called optically active materials.
• Dextrorotatory substances rotate the plane of polarization to the right (clockwise), while levorotatory substances rotate it to the left (counter-clockwise).
• Biot's laws state the angle of rotation depends on:
  • Length of the optically active substance traversed (θ ∝ l).
  • Concentration of the solution (θ ∝ c, where c = m/v, mass/volume).
  • Algebraic sum of rotations if multiple substances are present (θ = θ₁ + θ₂ + θ₃ + ...).
  • Inversely proportional to the square of the wavelength (θ ∝ 1/λ²).

Special Polarization Cases

• Circularly polarized light arises when two plane-polarized waves of equal amplitude are superimposed with a phase difference of π/2, 3π/2, 5π/2, etc.
• Equation of a circle (x² + y² = a²) shows circular polarization is achieved when a = b.

Specific Rotation

• Specific rotation is the rotation produced by one decimeter length of a solution containing one gram of optically active substance per cc, at a given wavelength and temperature.
• Represented by S = θ / (l × c), where θ is the rotation in degrees, l is the length in decimeters, and c is the concentration in g/cc.
• Specific rotation is typically expressed in units of degrees (dm)⁻¹ (g/cc)⁻¹
• Examples:
  • Cane sugar: +66.5° (dm)⁻¹ (g/cc)⁻¹
  • Glucose: +52° (dm)⁻¹ (g/cc)⁻¹
  • Fructose: -91° (dm)⁻¹ (g/cc)⁻¹

Polarimeter

• A polarimeter is an optical instrument used to measure the angle through which the plane of polarization is rotated by an optically active substance.
• When used specifically to determine the quantity of sugar in a solution, it is known as a saccharimeter.

Laurent's Half-Shade Polarimeter

• Laurent's half-shade polarimeter is used to find the optical rotation of certain solutions, and, specifically, is used to determine the concentration of sugar solutions.
• Known as a saccharimeter when ascertaining sugar concentration.
• Works because if specific rotation of sugar is known, the concentration can be determined.

Construction

• The setup includes a monochromatic light source (S), a Nicol prism polarizer (P), a half-shade device (H), a glass tube (T) containing the solution, an analyzing Nicol prism (A), and a telescope (E).
• Circular scale (C) is divided to measure the rotation.

Action of Half-Shade Device

• Consists of a semi-circular plate ADB of glass cemented to a semi-circular plate ACB of quartz.
• The quartz plate is a half wave plate.

Further Details on Polarimeter Function

• Light passes through the polarizer P and is incident normally on the half-shade plate, vibrating along OP.
• After passing through the glass, vibration remains along OP.
• After passing through quartz, the light splits into O and E components.
• Vibrations of both O and E are displaced in components along differing lines.

Analyzing the Fringes

• If the analyzing Nicol is fixed with its principal plane || to OP, the plane-polarized light through the glass half will pass and hence it will appear brighter than the quartz half from which light is partially obstructed.
• If the principal plane of the Nicol is || to OD, the quartz half will appear brighter than the glass half .

Matching the Halves

• When the principal plane of the analyzer is || to ADB, the two halves will appear equally bright, provided all aspects of intensity and vibration are accounted for.

Determining Sugar Solution

• Water is used to find the reading on the circular scale corresponding to darkness using equipment.
• Repeat after the equipment is filled with sugar.
• Compare the difference in reading for optical rotation, and factor in decimeter differences.

Concentration Calculation

• Formula of sugar is C = θ/lxs g/cc

Application

• Useful for industries, estimating sugar.
• Recalibrating makes the equipment a saccharimeter.