Comprehensive Study Notes on Geometrical and Wave Optics

Fundamental Concepts and Nature of Light

Light is a fundamental form of energy that allows for the perception and visualization of objects with the human eye. From a scientific perspective, light is categorized as an electromagnetic wave with a wavelength that falls within the visible portion of the electromagnetic spectrum, ranging from $400\,\text{nm}$ to $750\,\text{nm}$. The human eye perceives objects through two primary mechanisms: by the light the objects produce themselves or by the light they reflect from other sources. Objects that generate their own light are classified as luminous; prominent examples include the sun, electric light bulbs, and candle flames. In contrast, non-luminous objects do not produce light but are visible only when light from external sources falls upon them and is reflected into the eye. A notable example of a non-luminous object is the moon, which is visible at night not because it generates light, but because it reflects light originating from the sun.

Optics represents the branch of physics dedicated to the study of the behavior and properties of light. This scientific field encompasses the interaction between light and matter, as well as the design and construction of various instruments intended to use or detect light radiation. While optics primarily focuses on visible, ultraviolet, and infrared light, the principles often apply to other electromagnetic radiations—such as X-rays, microwaves, and radio waves—due to their shared wave-like nature. For the purposes of systematic study, optics is divided into two primary sub-disciplines: physical or wave optics, and geometrical or ray optics. Physical optics concerns itself with the wave nature of light and explains phenomena including interference and diffraction. Geometrical optics explores the formation of images through lenses and mirrors, relying on specific geometrical laws that govern light propagation.

Historical Theories of Light

Newton’s corpuscular theory of light posits that light is composed of excessively tiny, discrete particles known as corpuscles. According to this theory, these corpuscles are emitted from light sources and travel in straight lines at high velocities. When these particles enter the human eye, they strike the retina to produce the sensation of vision or a perceived image of an object. Newton further suggested that the varying colors of light are attributable to the differing sizes of these corpuscles.

Huygens’s wave theory of light, proposed by Christian Huygens, offers an alternative perspective by suggesting that light is a form of energy that propagates in the form of waves. This theory assumes that every point within a light source emits waves in all directions through a hypothetical, all-encompassing medium referred to as Ether. In this framework, a medium is strictly necessary for the propagation of light waves, and the entirety of space is filled with this imaginary Ether. Huygens also noted that these light waves possess extremely short wavelengths.

The quantum theory of light, introduced by Max Planck in 1905, established that energy radiated or absorbed by matter is not continuous but exists in discrete quantities. Energy cannot have fractional values; rather, it must be an integral multiple of a fixed quantity known as a quantum. These packets or bundles of energy are referred to as photons or quanta.

Principles of Geometrical Optics

Geometrical optics conceptualizes light propagation through the use of rays, which are approximate paths light follows under specific conditions. This framework relies on several fundamental assumptions regarding light behavior. Light rays propagate in straight lines when traveling through a homogeneous medium, a principle known as rectilinear propagation. When light encounters the interface between two dissimilar media, it may bend or split into two paths. Furthermore, light can follow curved paths if it passes through a medium where the refractive index changes continuously, and it may be absorbed or reflected upon striking glossy surfaces.

The study of geometrical optics is governed by three fundamental laws: the law of rectilinear propagation, the law of reflection, and the law of refraction. Collectively, these laws are encompassed by a general principle known as Fermat's principle of least time.

Fermat's Principle of Least Time

In 1658, Pierre De Fermat, a French mathematician, formulated the principle of least time. It states that when a light ray passes from one point to another through any number of media and via any number of reflections or refractions, it chooses the path for which the time taken is the minimum. However, subsequent observations showed that in certain scenarios, such as image formation by lenses, the time taken by light is not always a minimum; it can be a maximum or remain stationary (neither maximum nor minimum). Consequently, the principle was refined into Fermat's principle of stationary time or the principle of extreme path. This modified version states that a light ray chooses a path between two points such that the time taken is either minimum, maximum, or stationary.

Laws of Reflection and Fermat's Principle

Reflection occurs when a light ray strikes a smooth, polished surface separating two media and returns to the original medium. The surface where this occurs is called the reflecting surface. The first law of reflection states that the incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie within the same plane. This can be demonstrated by considering a point $M'$ on a mirror that is not on the same plane as the object point $P$ and the normal. Using the Pythagorean theorem, the path length through any point $M'$ outside the primary plane is always greater than the path through a point $M$ on the primary plane. Since Fermat's principle requires the shortest path, the light must travel along the path where the point of incidence $M$ lies in the same plane as the normal and the rays.

The second law of reflection dictates that the angle of incidence is equal to the angle of reflection ($i = r$). To derive this from Fermat’s principle, consider a path $PMP'$ with a total distance $s$. Let the vertical distances of points $P$ and $P'$ from a reflecting plane be $a$ and $b$, respectively, the horizontal distance between them be $c$, and the point of incidence be located at distance $x$ from the perpendicular projection of $P$. The total path length is:

$s = \sqrt{a^2 + x^2} + \sqrt{(c - x)^2 + b^2}$

Applying Fermat's principle, the derivative of $s$ with respect to $x$ must be zero for the path to be minimum:

$\frac{ds}{dx} = \frac{1}{2} \times \frac{2x}{\sqrt{a^2 + x^2}} - \frac{1}{2} \times \frac{2(c - x)}{\sqrt{(c - x)^2 + b^2}} = 0$

$\frac{x}{\sqrt{a^2 + x^2}} = \frac{c - x}{\sqrt{(c - x)^2 + b^2}}$

From the geometry of the system, these expressions represent $\sin(i)$ and $\sin(r)$. Therefore, $\sin(i) = \sin(r)$, leading to the conclusion that $i = r$. The second differential coefficient $\frac{d^2s}{dx^2}$ is positive, confirming that this stationary path is indeed a minimum.

Laws of Refraction and Snell's Law

Refraction is the phenomenon where a light ray bends, either toward or away from the normal, as it passes from one homogeneous medium to another. The first law of refraction states that the incident ray, the refracted ray, and the normal at the point of incidence all lie in one plane. Similar to reflection, Fermat's principle proves this because any path deviating from this plane would be longer and thus not the actual path taken by light.

The second law of refraction, known as Snell's law, specifies that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media. In optical terms, the product of the refractive index $\mu$ and the distance $S$ traveled in that medium is called the optical path. For light traveling from a point $P$ in a medium with refractive index $\mu_1$ to point $P'$ in a medium with index $\mu_2$, the total optical path is:

$S = \mu_1 \sqrt{a^2 + x^2} + \mu_2 \sqrt{(c - x)^2 + b^2}$

Setting $\frac{dS}{dx} = 0$ to find the minimum optical path:

$\frac{\mu_1 x}{\sqrt{a^2 + x^2}} - \frac{\mu_2 (c - x)}{\sqrt{(c - x)^2 + b^2}} = 0$

$\mu_1 \sin(i) = \mu_2 \sin(r)$

$\frac{\sin(i)}{\sin(r)} = \frac{\mu_2}{\mu_1} = {_1\mu_2}$

${_1\mu_2}$ represents the refractive index of the second medium with respect to the first. While this condition holds for plane surfaces, for curved surfaces the path might be a maximum or minimum.

Properties of the Refractive Index

Refractive index is a relative property of two media. The absolute refractive index is defined as the ratio of the speed of light in free space ($c$) to its speed in the medium ($v$):

$\mu = \frac{c}{v}$

Since frequency ($\nu$) remains constant during refraction, and $c = \nu \lambda$, the refractive index can also be expressed as the ratio of the wavelength in vacuum ($\lambda$) to the wavelength in the medium ($\lambda_m$):

$\mu = \frac{\lambda}{\lambda_m}$

The absolute refractive index depends on the nature of the medium, the wavelength of incident light, and the temperature. When light enters a denser medium from a rarer one ($\mu_2 > \mu_1$), it bends toward the normal ($i > r$). Conversely, when entering a rarer medium from a denser one ($\mu_2 < \mu_1$), it bends away from the normal ($i < r$). If the angle of incidence is zero ($i = 0$), the ray passes through undeviated independently of the refractive index, though its velocity still changes.

Total Internal Reflection

Total internal reflection (TIR) occurs when light travels from a denser medium toward a rarer medium. As the angle of incidence in the denser medium increases, the angle of refraction in the rarer medium also increases. The critical angle ($C$) is defined as the specific angle of incidence in the denser medium for which the angle of refraction is exactly $90^{\circ}$. If the angle of incidence exceeds this critical value, the light is not refracted but is reflected back into the denser medium.

The conditions for TIR are: first, the light must be moving from a denser to a rarer medium; second, the angle of incidence must be greater than the critical angle. The relationship between the refractive indices and the critical angle is derived from Snell's law:

$\frac{\sin(C)}{\sin(90^{\circ})} = \frac{\mu_r}{\mu_d}$

$\sin(C) = \frac{\mu_r}{\mu_d}$

If the rarer medium is air ($\mu_r = 1$), then $\sin(C) = \frac{1}{\mu_d}$.

Optical Fiber Technology

Optical fiber is a technology used for data transmission via light pulses traveling through a long fiber made of glass or plastic. These fibers utilize total internal reflection to propagate light down the length of the cable, even if the fiber is bent. Unlike metal wires, optical fibers are immune to electromagnetic interference and experience significantly lower signal damage and loss.

A standard optical fiber consists of a core, which is a solid dielectric cylinder with refractive index $n_1$, and a cladding, a surrounding material with a lower refractive index $n_2$. The cladding serves several purposes: reducing scattering losses, adding mechanical strength, and protecting the core from surface contaminants. Optical fibers are classified into several types. Based on refractive index, they are either Step Index Fibers (uniform index in the core) or Graded Index Fibers (refractive index decreases as radial distance from the axis increases). Based on material, they are Plastic Optical Fibers (PMMA core) or Glass Fibers. Based on propagation mode, they are Single-Mode Fibers (for long distances) or Multimode Fibers (for shorter distances).

A fiber optic relay system consists of a transmitter (to encode light signals), the optical fiber (the medium), an optical receiver (to decode the signals), and optical regenerators (necessary for maintaining signals over long distances). Physical advantages of these cables include high capacity, small size, lightweight, lack of spark hazards, resistance to corrosion, and lower raw material costs. However, they are fragile without protection and require high installation costs and multiple repeaters over long distances. Applications include telephone systems, submarine cables, CATV systems, CCTV, and emergency services.

Interference of Light and Young’s Double Slit Experiment

Interference is the phenomenon where two coherent waves superpose to form a resultant wave of higher or lower amplitude. For a stable interference pattern, the sources must be coherent (constant phase difference) and monochromatic (single wavelength). Constructive interference happens when the crest of one wave meets the crest of another (or trough meets trough), resulting in a larger amplitude. Destructive interference happens when the crest of one wave meets the trough of another, resulting in zero amplitude.

In 1801, Thomas Young demonstrated this with his Double Slit Experiment. Light from a source $S_0$ passes through two equidistant pinholes $S_1$ and $S_2$. The resulting spherical waves overlap, creating alternating bright and dark bands (fringes) on a screen. Using two independent sources is impossible for this experiment because they cannot maintain the necessary constant phase difference.

The theory of interference fringes explains the distribution of intensity. Let $d$ be the distance between the two coherent sources and $D$ be the distance between the sources and the screen. For a point $P$ at distance $x$ from the center $C$, the path difference is calculated using the Pythagorean theorem for the two paths $S_2P$ and $S_1P$. After simplification based on the assumption that $d \ll D$:

$\text{Path Difference} = \frac{x \times d}{D}$

Bright fringes occur when the path difference is an integral multiple of the wavelength ($n \lambda$):

$x = \frac{n \times \lambda \times D}{d}$

Dark fringes occur when the path difference is an odd multiple of half-wavelength ($(2n+1)\frac{\lambda}{2}$):

$x = (2n + 1) \times \frac{\lambda \times D}{2d}$

The fringe width (the distance between two consecutive bright or dark fringes) is given by:

$\beta = \frac{\lambda \times D}{d}$

Thus, fringe width increases with longer wavelengths, greater distance to the screen, or by bringing the two slits closer together.

Newton’s Rings Experiment

Newton’s rings are an interference pattern created by the reflection of light between a spherical surface (a plano-convex lens) and an adjacent flat glass plate. This setup creates a thin film of air whose thickness is zero at the point of contact and increases outward. When illuminated with monochromatic light, concentric circular dark and bright fringes are formed. The optical path difference $\Delta$ for normal incidence is:

$\Delta = 2t - \frac{\lambda}{2}$

Dark rings occur when $2t = m \lambda$. Using the radius of curvature $R$ of the lens and the radius of the ring $r$, geometry dictates $r^2 = 2Rt$. Substituting for $t$ and using diameter $D = 2r$, the diameter of the $m^{th}$ dark ring is:

$D_m^2 = 4mR \lambda$

To find the wavelength of light $\lambda$, the diameters of two rings ($m^{th}$ and $(m+n)^{th}$) are measured:

$\lambda = \frac{D_{m+n}^2 - D_m^2}{4nR}$

Polarization of Light

Polarization is the process of restricting light wave vibrations to a single plane. Sound waves (longitudinal) do not exhibit polarization, but light (transverse) does. Unpolarized light vibrates in all planes normal to the direction of propagation, while polarized light vibrates in a specific plane. The plane containing the vibrations is the plane of vibration, and the plane perpendicular to it is the plane of polarization.

Polarization takes three forms: Linear, Circular, and Elliptical. Linearly polarized light is a plane wave where the electric field oscillates in one direction. Circularly polarized light occurs when two perpendicular waves of equal amplitude differ in phase by $90^{\circ}$, causing the electric field vector to rotate in a circle. If rotating counterclockwise from the receiver's perspective, it is right-circularly polarized; clockwise is left-circularly polarized. Elliptical polarization occurs when the amplitudes are unequal or the phase difference is not $90^{\circ}$.

Linear polarization can be produced via reflection, refraction, scattering, selective absorption, and double refraction. Brewster’s Law states that the tangent of the polarizing angle $\theta_p$ is equal to the refractive index of the medium:

$\mu = \tan(\theta_p)$

This occurs when the reflected ray and refracted ray are at right angles ($90^{\circ}$). Malus's Law describes the intensity $I$ transmitted through an analyzer:

$I = E \times \cos^2(\theta)$

where $E$ is the incident intensity and $\theta$ is the angle between the transmission axes of the polarizer and analyzer. Specific rotation $S$ for optically active substances is defined by:

$[S]^t = \frac{10\theta}{l \times C}$

where $\theta$ is the rotation in degrees, $l$ is the length in decimeters (or centimeters in the alternate formula variant), and $C$ is the concentration.

Diffraction of Light

Diffraction is the slight bending of light as it passes around the edge of an object or through an aperture. Significant bending occurs if the wavelength of light is comparable to the size of the opening. It causes the "silver lining" effect on clouds. Conditions for visible diffraction include having a very sharp edge or a hole with a diameter on the order of the wavelength of light.

There are two classes of diffraction. Fresnel diffraction occurs when the light source and screen are at finite distances from the obstacle; the wavefronts are spherical or cylindrical. Fraunhofer diffraction occurs when the source and screen are effectively at an infinite distance (achieved with lenses), and the incident wavefronts are plane.

Questions & Discussion

Self Assessment Questions (SAQ)

1. What is total internal reflection? Total internal reflection is the phenomenon where a light ray traveling from a denser medium to a rarer medium is reflected back into the denser medium because its angle of incidence exceeds the critical angle.

2. What is the critical angle for a medium of refractive index $\sqrt{2}$? Given $\mu_d = \sqrt{2}$ and assuming the rarer medium is air ($\mu_r = 1$), the relationship is $\sin(C) = \frac{1}{\mu_d}$. Thus, $\sin(C) = \frac{1}{\sqrt{2}}$, which means $C = 45^{\circ}$.

3. Using Fermat principle, establish the condition of total internal reflection. As established via Fermat’s principle in the laws of refraction, $\mu_1 \sin(i) = \mu_2 \sin(r)$. For total internal reflection to occur from a medium $\mu_1$ to air ($\mu_2 = 1$), the maximum possible refraction angle is $90^{\circ}$, setting $\sin(i) = \frac{1}{\mu_1}$. Any increase in the optical path beyond this point through the second medium results in the ray remaining in the first medium to satisfy the stationary time requirement for the path taken.