Comprehensive Study Guide on Light, Wave Optics, and Lens Systems of Lenses, and Diffraction

Fundamental Properties of Light

Wave Nature: Light travels from one point to another in the form of waves. These waves are characterized by a short wavelength and a high frequency.
Velocity: The velocity of light in a vacuum is defined as $3.0 \times 10^8\,\text{m/s}$ .
Wave Behaviors: Like other wave types, light can be reflected, refracted, and diffracted.
Transverse Nature: Because light is a transverse wave, it can also be polarised.

Huygens' Construction for a New Wavefront

Wavefront Definition: A wavefront is considered to be composed of many individual source points.
Secondary Centres of Disturbance: According to Huygens, every point on a wavefront becomes a new or secondary center of disturbance for the purpose of obtaining the position of a new wavefront after a further time $t$ .
Resultant Wavefront: The new wavefront is defined as the tangent to all the arcs drawn from these secondary centers.
Geometric Relation: The relation between wavefronts and the direction of the wave (represented by a ray) remains constant: they are perpendicular to one another at all points.

Huygens' Construction for Reflection at a Boundary

Parallel Rays and Wavefronts: Suppose a beam of parallel rays between $HA$ and $LC$ is incident on a plane mirror. A plane wavefront $AB$ is normal to these rays as it reaches the mirror surface.
Point of Disturbance: At the instant the wavefront hits the mirror, point $A$ begins to act as a center of disturbance.
Wavefront Propagation: To determine the new wavefront at the instant the disturbance at $B$ reaches $C$ , we note that the wave originally at $A$ travels to $D$ . Thus, the reflected wavefront at that instant is shown by the line $DC$ .
Reflection Symmetry: Similarly, a disturbance reaching point $P$ will reflect toward point $M$ (instead of continuing toward $N$ ).
Law of Reflection: Based on this construction, it is observed that the angle between the incident ray and the normal is exactly equal to the angle between the reflected ray and the normal.

Reflection at Plane Surfaces

Reflectivity of Surfaces: Highly polished metal surfaces are capable of reflecting between $80\%$ to $90\%$ of the incident light.
Mirror Construction: Mirrors in daily use are typically manufactured by depositing silver onto the back of a glass sheet.
Angle of Incidence ( $i$ ): When a ray of light is incident on a plane mirror, the angle the incident ray makes with the normal to the mirror is the angle of incidence.
Angle of Reflection ( $r$ ): The angle made by the reflected ray with the normal is the angle of reflection.
Laws of Reflection: - The reflected ray, the incident ray, and the normal to the mirror at the point of incidence all lie within the same plane. - The angle of incidence is equal to the angle of reflection ( $i = r$ ).

Images in Plane Mirrors

Image Formation: If a point object $A$ is placed in front of a mirror, a ray $OA$ incident on the mirror reflects along $AX$ such that $\angle OAN = \angle NAX$ (where $AN$ is the normal). A ray $OB$ incident on the mirror reflects back.
Virtual Image ( $I$ ): The reflected rays appear to originate from a point $I$ behind the mirror, where $I$ is the intersection point of the reflected rays. This image is "virtual" because the rays do not actually exist at that point.
Distance Properties: - The object and image are at equal perpendicular distances from the mirror surface. - For any point on the mirror, the distance to the object ( $AO$ ) is equal to the distance to the image ( $AI$ ).
Lateral Inversion: The left side of an object appears on the right side of the image and vice versa. - Example: Consider an E-shaped object. The image of point $a$ is at $a'$ at an equal distance behind the mirror. Point $b$ on the left of $a$ appears at $b'$ on the right of $a'$ .

Refraction at a Plane Surface

Wavefront Transition: Consider parallel rays incident from air onto a water medium. As a plane wavefront $AB$ reaches the surface, each point between $A$ and $B$ becomes a new center of disturbance. The wavefront changes direction as it enters the liquid.
Velocity and Distance Relations: - Let $t$ be the time taken to travel from $B$ to $D$ . The distance $BD = v_{a} \times t$ , where $v_{a}$ is the velocity of light in air. - In the same time $t$ , the disturbance from $A$ travels a distance $AC$ in water where $AC = v_{w} \times t$ , and $v_{w}$ is the velocity of light in water.
Derivation of Refractive Index: - $\sin(\theta_{a}) = \frac{BD}{AD} = \frac{v_{a} \times t}{AD}$ - $\sin(\theta_{g}) = \frac{AC}{AD} = \frac{v_{g} \times t}{AD}$ - Dividing the equations: $\frac{\sin(\theta_{a})}{\sin(\theta_{g})} = \frac{v_{a}}{v_{g}} = n$ , where $n$ is the refractive index.

Laws of Refraction and Snell's Law

General Behavior: When light hits a glass surface at point $O$ , part is reflected (per the laws of reflection) while the rest enters the glass and changes direction (refraction).
Angle of Refraction ( $r$ ): The angle made by the refracted ray with the normal at point $O$ .
Definition of Laws: - The incident ray, refracted ray, and normal at the point of incidence all lie in the same plane. - Snell's Law: For two given media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant: $\frac{\sin(i)}{\sin(r)} = n$ .
Wavelength Dependency: The refractive index depends on the wavelength of light. Values are typically given for yellow light. - Longer wavelengths have slightly smaller refractive indices. - Shorter wavelengths have larger refractive indices. - This explains why red light bends the least and violet light bends the most.
Refractive Index Formula: $n_{medium} = \frac{c_{vacuum}}{c_{medium}}$ .
General Relation: $\frac{\sin(\theta_{1})}{\sin(\theta_{2})} = \frac{v_{1}}{v_{2}} = \frac{n_{2}}{n_{1}}$ , leading to $n_{1} \sin(\theta_{1}) = n_{2} \sin(\theta_{2})$ .
Calculation Example: A ray incident on a glass-to-water boundary at $50^\circ$ to the normal. $n_{glass} = 1.5$ , $n_{water} = 1.33$ . - $1.50 \sin(50^\circ) = 1.33 \sin(r)$ - $r = 59.8^\circ$

Real and Apparent Depth/Thickness

Phenomenon: The bottom of a pool appears shallower than it is due to refraction.
Mathematical Relation: Consider an object $O$ below the surface. A ray is refracted into air away from the normal. An observer sees the image at $I$ . - $n_{water} = \frac{OB}{IB} = \frac{\text{Real depth}}{\text{Apparent depth}}$
Application to Solids: The formula applies to thickness: $n_{glass} = \frac{\text{Real thickness}}{\text{Apparent thickness}}$ . - Example: A glass block is $8\,\text{cm}$ thick with $n_{glass} = 1.5$ . - $\text{Apparent thickness} = \frac{8}{1.5} = 5.33\,\text{cm}$ . - The image shifts $1.67\,\text{cm}$ toward the observer.

Total Internal Reflection (TIR)

Transition from Dense to Rare: If a ray moves from glass toward air at a small angle $i$ , most light is refracted away from the normal (bright ray) and a little is reflected (weak ray).
Increasing Angle of Incidence: As $i$ increases, the angle of emergence increases.
Critical Angle ( $c$ ): The specific angle of incidence where the refracted ray travels along the boundary (angle of refraction = $90^\circ$ ). - Formula: $n_{glass} \sin(c) = n_{air} \sin(90^\circ)$ . - $\sin(c) = \frac{1}{n_{glass}}$ .
TIR Definition: If the angle of incidence exceeds the critical angle, all light is reflected back into the dense medium; no refraction occurs.
Material Specifics: - Glass ( $n=1.51$ for yellow light): Critical angle $c = 41.5^\circ$ . - Blue light has a smaller critical angle than red light because it has a higher refractive index. - Transition from glass ( $n=1.51$ ) to water ( $n=1.33$ ): $1.51 \sin(c) = 1.33 \sin(90^\circ)$ , yielding $c = 63^\circ$ .

Applications and Dispersion

Fibre Optics: A core and cladding with a small difference in refractive index result in a large critical angle. Only light within the "acceptance cone" is transmitted.
Dispersion of White Light: Caused by differences in speeds of various colors within a medium. - Red light has a smaller refractive index; blue has a larger index. - At a glass surface, while light travels from $C$ to $D$ in air, red light moves to $R$ and blue light moves to $B$ . Blue is refracted more because its speed is lower than white/red in glass.

Converging and Diverging Lenses

Definitions: - Converging (Convex): Thicker in the middle than at the edges. Converges parallel rays to a real focal point $F$ . - Diverging (Concave): Thinner in the middle than at the edges. Diverges parallel rays outward from a virtual focal point $F$ .
Principal Axis: The line joining the centers of curvature of the two surfaces, passing through the middle of the lens.
Sign Convention for Focal Length ( $f$ ): - Converging lens: Real focus, $f$ is positive (+). - Diverging lens: Virtual focus, $f$ is negative (-).
Thin Lens Approximation: A lens can be viewed as multiple prisms placed together.
Image Characteristics in Converging Lenses: - Object at infinity: Image at focus, real, and inverted. - Object distance > $f$ : Image is real and inverted. Magnification increases as the object approaches $f$ . - Object distance < $f$ : Image is erect, magnified, and virtual. This is the "magnifying glass" application.
Image Characteristics in Diverging Lenses: Real objects always produce virtual, erect, and diminished images.

The Lens Equation and Magnification

Formula: $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$ - $u$ = object distance, $v$ = image distance, $f$ = focal length.
Sign Rules: - Real object/image: Positive (+). - Virtual object/image: Negative (-). - Converging lens $f$ : Positive (+). - Diverging (concave) lens $f$ : Negative (-).
Magnification ( $m$ ): - $m = \frac{\text{height of image}}{\text{height of object}} = \frac{v}{u}$ . - Signs of $v$ and $u$ are ignored for magnification ratios.

Interference and Coherent Sources

Constructive Interference: Waves arrive "in phase," resulting in higher amplitude (bright light).
Destructive Interference: Waves arrive in "antiphase," resulting in cancellation (dark spot or fringe).
Coherent Sources: Sources that emit waves of the same wavelength with a constant phase difference.
Creating Coherence: Since separate lamps have random atom emission resulting in varying phase differences, a single source is usually split (e.g., via two slits) to create two coherent sources.
Mathematical Conditions: - Bright fringe (Constructive): Path difference $BZ - AZ = m\lambda$ (where $m$ is a whole number). - Dark fringe (Destructive): Path difference is a half-wavelength ( $0.5\lambda$ , $1.5\lambda$ , etc.).

Young's Two-Slit Experiment

Setup: Monochromatic light from a single slit falls on two closely spaced slits. A screen shows bright and dark bands on either side of the central maximum.
Fringe Separation Formula: $\text{fringe separation} = \frac{\lambda \times D}{d}$ - $\lambda$ = wavelength. - $D$ = distance from slits to screen. - $d$ = distance between the two slits.
Experimental Adjustments: Increasing slit separation $d$ results in smaller fringe separation. Using longer wavelengths increases fringe separation.
Measuring $\lambda$ : In a lab, $d$ is typically $0.5\,\text{mm}$ and $D \approx 1.5\,\text{m}$ . One measures the distance across 10 fringes then divides by 10 to find average separation.

Diffraction Patterns

Definition: Interference occurring at a single opening or circular aperture, treated as having multiple point sources.
Single Slit: The first dark fringe occurs at an angle $\theta$ where $\sin(\theta) = \frac{\lambda}{a}$ ( $a =$ aperture width).
Resolution and Apertures: Increasing aperture width or decreasing wavelength results in a narrower diffraction pattern (higher resolution).
Telescope Objective: Forms a diffraction pattern of a star. For a circular objective: $\sin(\theta) = 1.22 \frac{\lambda}{diameter}$ .
Rayleigh Criterion: Two stars are resolved if the maximum intensity of the central pattern of one star falls on the edge of the central pattern of the other. If bright fringes overlap, they are unresolved.

Diffraction Gratings

Definition: A large number of parallel, equidistant slits ruled on metal or glass.
Angular Positions: $d \sin(\theta) = n\lambda$ - $d$ = distance between slits. - $n$ = order number (1, 2, 3, etc.).
Image Orders: - First order ( $n=1$ ). - Second order ( $n=2$ ). - Orders higher than those where \sin(\theta) > 1 are impossible.
Example: Grating with $600\,\text{lines/mm}$ . $d = \frac{1\,\text{mm}}{600} = 1.667 \times 10^{-6}\,\text{m}$ . - For yellow light ( $\lambda = 5.89 \times 10^{-7}\,\text{m}$ ), first order is at $20.7^\circ$ , second order at $45^\circ$ .
Measurement: Wavelength can be measured using a diffraction grating with a spectrometer by measuring the angle $2\theta$ between symmetric first-order images.
Spectra Comparison: - In a prism, red deviates least and violet deviates most. - In a grating ( $d \sin(\theta) = n\lambda$ ), violet (shorter $\lambda$ ) has a smaller angle $\theta$ than red; violet is deviated least in each order.

Polarization of Light Waves

Polaroid: An artificial crystalline material in thin sheets that allows only a particular polarization of light vibrations to pass.
Polaroid Experiment: - Two polaroids with parallel axes: Light passes through (slightly darker). - Rotating one polaroid: Light dims and disappears when axes are perpendicular. - Re-rotating: Light reappears and becomes brightest when parallel again.
Conclusion: This experiment proves light waves are transverse vibrations.
Vibration Mechanics: Unpolarized light has vibrations in many directions. When incident on a polaroid, only components parallel to the polaroid axis pass through; perpendicular components are absorbed. This creates plane polarized light.

Problems and Quantitative Applications

Refraction Problem: Light incident at $25^\circ$ from glass ( $n=1.5$ ) to water ( $n=1.3$ ). - Solve for angle of refraction: $1.5 \sin(25^\circ) = 1.3 \sin(r)$ . - Wavelength calculation: Air $\lambda = 500\,\text{nm}$ . Since $n = \frac{\lambda_{air}}{\lambda_{medium}}$ , find glass and water wavelengths.
Lens Problems: - Determining magnification ( $m = \frac{h_{i}}{h_{o}}$ ) and focal length from image distance. - Object at $15\,\text{cm}$ , $f = 10\,\text{cm}$ . Image distance $v$ and magnification calculation. - Convex lens used as magnifying glass to produce image $4\times$ larger; find $u$ .
Sound Wave Refraction: Comparison of sound wave transmission/reflection at an air-water boundary ( $v_{air} = 340\,\text{m/s}$ , $v_{water} = 1500\,\text{m/s}$ ).
Fibre Optic Problem: Core ( $n=1.52$ ), Cladding ( $n=1.48$ ). Determine critical angle and maximum acceptance angle.
Spectra Separation: Sodium lines ( $\lambda_{1} = 5.890 \times 10^{-7}\,\text{m}$ , $\lambda_{2} = 5.896 \times 10^{-7}\,\text{m}$ ) with grating ( $1.0 \times 10^6\,\text{lines/m}$ ). Find angular and linear separation at $1.0\,\text{m}$ .