Polarized Slide

PHY-1109 Physics

Polarization Notes (LM2) Polarization PYQ Questions Polarization Topic Q&A Polarization Notes (V1) Polarization Notes (V2) Polarized Slide

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I. Polarization by Reflection

Discovery and Phenomenon

Polarization of light by reflection from a glass surface was discovered by Malus in 1808. Polarized light is obtained when ordinary light is reflected by a plane sheet of glass.

If light is incident along path AB on a glass surface, reflected along BC, and a tourmaline crystal is placed in the path BC and rotated slowly, the light will be completely extinguished only at one particular angle of incidence. This specific angle of incidence is known as the polarizing angle. For a glass surface, this angle is equal to $57.5^{\circ }$. Polarized light can also be produced by reflection from a water surface.

II. Brewster’s Law

In 1811, Brewster performed experiments to study the polarization of light by reflection at the surfaces of different media.

The Law

Brewster proved that the tangent of the angle of polarization is numerically equal to the refractive index ($\mu$) of the medium. Moreover, when light is incident at this polarizing angle, the reflected ray (BC) and the refracted ray (BD) are perpendicular to each other.

Derivation

Suppose unpolarized light is incident at an angle $i$ (equal to the polarizing angle) on the glass surface. It is reflected along BC and refracted along BD.

1. From Snell’s law: $\mu =\frac{\sin i}{\sin r}(1)$

2. From Brewster’s law: $\mu =\tan i=\frac{\sin i}{\cos i}(2)$

3. Comparing (1) and (2): $\cos i=\sin r=\cos \left(\frac{\pi }{2}-r\right)$ $\therefore i=\frac{\pi }{2}-r$

4. Result: $i+r=\frac{\pi }{2}$ This also means the angle between the reflected and refracted rays ($\angle CBD$) is equal to $\frac{\pi }{2}$ (or $90^{\circ }$).

(See Source and for diagram showing Brewster’s setup.)

III. Malus’ Law

Definition

When a beam of light, polarized by reflection at one plane surface, is allowed to fall on a second plane surface at the polarizing angle, the intensity of the twice reflected beam varies with the angle between the planes of the two surfaces.

The law of Malus states that the intensity of the polarized light transmitted through the analyzer varies as the square of the cosine of the angle ($\theta$) between the plane of transmission of the analyzer and the plane of the polarizer.

Derivation

Let $a$ be the amplitude of the vibrations transmitted or reflected by the polarizer (OP). $\theta$ is the angle between the planes of the polarizer and the analyzer.

The amplitude OP can be resolved into two components: (i) $a\cos \theta$ along OA. (ii) $a\sin \theta$ along OB.

Only the $a\cos \theta$ component is transmitted through the analyzer.

The intensity of the transmitted light ($E_1$) through the analyzer is proportional to the square of the amplitude: $E_1=(a\cos \theta {)}^2=a^2{\cos }^2\theta$ But $E=a^2$, where $E$ is the intensity of the incident polarized light. $\therefore E_1=E{\cos }^2\theta$ or $E_1\propto {\cos }^2\theta$

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IV. Double Refraction

Discovery and Calcite

Erasmus Bartholinus discovered in 1669 that when light is refracted by a crystal of calcite, it gives two refracted rays. This phenomenon is called double refraction.

Calcite or Iceland spar (${\text{CaCO}}_3$) is crystallized calcium carbonate found in large quantities in Iceland as very large transparent crystals. It crystallizes in many forms and can be reduced by cleavage or breakage into a rhombohedron bounded by six parallelograms. The angles of the rhombohedron are equal to $102^{\circ }$ and $78^{\circ }$ (more accurately $101^{\circ }55^{\prime }$ and $78^{\circ }{5}^{\prime }$).

Ordinary and Extraordinary Rays

When a calcite crystal is placed over an ink dot on paper, two images will be observed. If the crystal is rotated slowly, one image remains stationary, and the second image rotates.

• The stationary image is known as the ordinary image.
• The second (rotating) image is known as the extraordinary image.

The phenomenon of double refraction is absent when light is allowed to enter the crystal along the optic axis.

Refractive Indices in Calcite

The ordinary ray has a refractive index: ${\mu }_o=\frac{\sin i}{\sin r_1}$ The extraordinary ray has a refractive index: ${\mu }_e=\frac{\sin i}{\sin r_2}$

In calcite, ${\mu }_o>{\mu }_e$ because $r_1$ is less than $r_2$. Therefore, the velocity of light for the ordinary ray inside the crystal is less compared to the velocity of light for the extraordinary ray.

Principal Section and Principal Plane

1. Principal Section of the Crystal: A plane which contains the optic axis and is perpendicular to the opposite faces of the crystal is called the principal section of the crystal. Since a crystal has six faces, there are three principal sections for every point. A principal section cuts the surface of a calcite crystal in a parallelogram with angles $109^{\circ }$ and $71^{\circ }$.

2. Principal Plane: A plane in the crystal drawn through the optic axis and the ordinary ray is defined as the principal plane of the ordinary ray. Similarly, a plane in the crystal drawn through the optic axis and the extraordinary ray is defined as the principal plane of the extraordinary ray. In general, the two planes do not coincide. However, if the plane of incidence is a principal section, then the principal section of the crystal and the principal planes of the ordinary and extraordinary rays coincide.

V. Nicol Prism

Description and Construction

The Nicol prism is an optical device used for producing and analyzing plane polarized light. It was invented by William Nicol in 1828.

1. Initial Shape: A calcite crystal is taken whose length is three times its breadth. A’BCDEFG’H represents such a crystal, having A’ and G’ as its blunt corners. One of the principal sections has an angle A’CG’ $=71^{\circ }$.
2. Modification: The faces A’BCD and EFG’H are grounded so that the angle ACG becomes $68^{\circ }$ instead of $71^{\circ }$.
3. Cementing: The crystal is then cut along the plane AKGL. The two cut surfaces are polished optically flat and cemented together by Canada balsam.
4. Refractive Indices:
   • Refractive index for the ordinary ray (${\mu }_o$) $=1.658$.
   • Refractive index for Canada balsam (${\mu }_B$) $=1.550$.
   • Refractive index for the extraordinary ray (${\mu }_e$) $=1.486$. (Note that the refractive index of Canada balsam lies between the refractive indices for the ordinary and extraordinary rays for calcite.)

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Function (Total Internal Reflection)

The function relies on the difference in refractive index to cause total internal reflection of the Ordinary ray (O-ray).

The refractive index for the ordinary ray with respect to Canada balsam is: $\mu =\frac{1.658}{1.550}$ The critical angle (${\theta }_c$) calculation relies on: $\sin \theta =\frac{1}{\mu }$ The critical angle is calculated as: $\therefore \theta =69^{\circ }$

Nicol Prism as an Analyzer

The Nicol prism can be used for the production and detection of plane polarized light.

VI. Optical Activity and Specific Rotation

Optical Activity

When a polarizer and an analyzer are crossed (no light emerges out of the analyzer), if a quartz plate (cut with its faces parallel to the optic axis) is introduced between them, the light emerges out of the analyzer (${\text{N}}_2$).

The quartz plate turns the plane of vibration. As the plane polarized light enters the quartz plate, its plane of vibration is gradually rotated.

The action of turning the plane of vibration occurs inside the body of the plate and not on its surface. This phenomenon, or the property of rotating the plane of vibration by certain crystals or substances, is known as optical activity. The substance itself is known as an optically active substance.

The amount of rotation depends upon the thickness of the quartz plate and the wavelength of light.

Classification of Active Substances

1. Dextrorotatory (Right-handed): Substances that rotate the plane of vibration to the right. Right-handed rotation means that when the observer is looking towards the light traveling towards him, the plane of vibration is rotated in a clockwise direction.
2. Laevorotatory (Left-handed): Substances that rotate the plane of vibration to the left (anti-clockwise from the point of view of the observer).

Specific Rotation

Specific rotation ($S_{\lambda }^t$) is defined as the rotation produced by a decimeter (10 cm) long column of the liquid containing 1 gm of the active substance in one cc of the solution.

Formula: $S_{\lambda }^t=\frac{10\theta }{lc}$ (Where $\theta$ is the total rotation, $l$ is the length in decimeters, and $c$ is the concentration.)

summary:

Table 1: Key Polarization Phenomena & Laws

Concept	Core Principle	Key Relationship/Value
Polarization by Reflection	Polarized light is obtained when ordinary light reflects off a plane sheet of glass.	Polarizing angle for a glass surface is 57.5∘.
Brewster’s Law	The tangent of the angle of polarization (tani) is numerically equal to the refractive index (μ). Reflected and refracted rays are perpendicular.	μ=tani; Result: i+r=2π​.
Malus’ Law	The intensity (E1​) of polarized light transmitted through an analyzer varies as the square of the cosine of the angle (θ) between the polarizer and the analyzer.	Formula: E1​=Ecos2θ.

Table 2: Double Refraction and Calcite Properties

Property	Description/Characteristics	Value/State	Citation
Double Refraction	Phenomenon where light, when refracted by a crystal of calcite, produces two refracted rays.	Discovered by Erasmus Bartholinus in 1669.
Calcite	Also known as Iceland spar (CaCO3​).	Crystallizes as a rhombohedron with angles 102∘ and 78∘ (or 101∘55′ and 78∘5′).
Rays/Images	Ordinary Image (stationary) and Extraordinary Image (rotates).	Double refraction is absent along the optic axis.
Refractive Indices (\mu)	The ordinary ray index (μo​) is greater than the extraordinary ray index (μe​).	Velocity Result: Velocity of O-ray is less than the velocity of E-ray.
Principal Section	A plane containing the optic axis and perpendicular to opposite faces.	Cuts the surface in a parallelogram with angles 109∘ and 71∘.

Table 3: Nicol Prism and Optical Activity

Component	Description/Function	Key Values/Formula	Citation
Nicol Prism	Optical device used for producing and analyzing plane polarized light.	Invented by William Nicol in 1828.
Construction Detail	Calcite crystal is cut and cemented with Canada balsam. Faces are grounded so that the principal angle becomes 68∘ instead of 71∘.	μo​=1.658; μBalsam​=1.550; μe​=1.486.
O-Ray Elimination	The Ordinary ray undergoes total internal reflection.	Critical angle θ=69∘.
Optical Activity	The property of rotating the plane of vibration (like by a quartz plate).	Amount of rotation depends on thickness and wavelength of light.
Types of Rotation	Dextrorotatory (right-handed/clockwise) and Laevorotatory (left-handed/anti-clockwise).	Right-handed rotation is clockwise when observing light traveling toward the observer.
Specific Rotation (Sλt​)	Rotation produced by a 1 decimeter (10 cm) column of liquid containing 1 gm of active substance in 1 cc of solution.	Formula: Sλt​=lc10θ​.