Engineering Physics Notes - Wave Optics and Interference

Course Overview & Syllabus Details

• Course Title: Engineering Physics

• Course Code: 316U06C102

• Institution: KJ Somaiya School of Engineering (formerly K J Somaiya College of Engineering), Somaiya Vidyavihar University

Teaching and Examination Scheme

• Teaching Scheme:

  • Theory (TH): $02$

  • Practical (PRACT): $-$

  • Tutorial (TUT): $-$

  • Total Hours: $02$

• Credits Assigned:

  • Theory: $02$

  • Total Credits: $02$

• Examination Scheme & Marks Breakdown:

  • Continuous Assessment (CA):

  • Internal Assessment (IA): $20$

  • Mid Semester Exam (MSE): $30$

  • End Semester Exam (ESE) (Theory/On-Screen): $50$

  • Practical / Oral (PR/OR) Exam: $-$

  • Lab / Tutorial CA: $-$

  • Total Marks: $100$

Syllabus Overview by Modules

• Module 1: Wave Optics ($08$

  • Unit 1.1: Interference: Introduction, methods of interference, derivation of interference in thin parallel films, theoretical discussion of a wedge-shaped film and Newton's Ring.

  • Unit 1.2: Diffraction: Introduction, theoretical discussion of diffraction grating and resolving power ($07$

• Module 2: Photonics ($07$

  • Unit 2.1: Lasers: Interaction of radiation with matter, basic principles and components required for lasing, relation between Einstein's coefficients, threshold condition of laser.

  • Unit 2.2: Fiber Optics: Concept of total internal reflection, types of optical fiber, derivation of numerical aperture for step index fiber, important parameters related to an optical fiber, losses in optical fibers.

• Module 3: Quantum Mechanics-I ($07$

  • Unit 3.1: De-Broglie hypothesis and its various forms.

  • Unit 3.2: Heisenberg's uncertainty principle, its implications, and application.

  • Unit 3.3: Wavefunction and its probabilistic interpretation, condition of normalization.

• Module 4: Basics of Semiconductors and Electromagnetism ($08$

  • Unit 4.1: Intrinsic and Extrinsic semiconductors, mobility, conductivity in semiconductors.

  • Unit 4.2: Drift and Diffusion current densities.

  • Unit 4.3: Fermi-Dirac distribution function, position of Fermi Level in Intrinsic and Extrinsic Semiconductors.

  • Unit 4.4: Del operator, Gradient, divergence, curl, fundamental theorems and their physical interpretation.

Role of Physics in Engineering Disciplines

• Core Foundation: Physics is the underlying foundation of all engineering. Every device, machine, or system—such as smartphones, bridges, automobiles, and AI hardware—operates based on physical principles.

• Field-Specific Integrations:

  • Mechanical Engineering: Governed by Newton's Laws of Motion and Thermodynamics.

  • Electrical and Electronics Engineering: Built upon circuit theory, magnetism, and electromagnetic (EM) wave propagation.

  • Civil Engineering: Dependent on statics, dynamics, and fluid flow mechanics.

  • Computer and IT Engineering: Driven by semiconductor physics, optics, and electromagnetic theory.

  • Aerospace Engineering: Relies on motion laws, pressure distributions, and aerodynamics.

• Real-World Technological Applications:

  • Structural Stability (Bridges): Analyzed through mechanical equilibrium, forces, and torque.

  • Wireless Power Transmission: Operates via electromagnetic induction.

  • Aeronautical Flight: Explained by fluid dynamics and Bernoulli's Principle.

  • Electronics Thermal Management: Involves heat transfer mechanisms and electrical resistance.

  • Autonomous Vehicle Perception: Uses wave optics, lidar, and optical sensors.

• Perspective: As expressed by James Kip Finch: "The engineer has been and is a maker of history."

Module 1.1: Fundamentals of Interference

Definition and Principles

• Wave Interference: A wave phenomenon occurring when two or more coherent waves travel through the same medium simultaneously and overlap at a common spatial point.

• Principle of Superposition: When waves coincide, their resultant wave displacement or amplitude at any given instant in time and point in space equals the algebraic sum of the individual wave amplitudes.

Primary Classification of Interference

• Constructive Interference:

  • Occurs when interfering waves arrive in phase with each other (crests align with crests, troughs align with troughs).

  • Phase difference: $\tan(\theta) = 2\theta\frac{\theta}{2} = 2n\theta$

  • Path difference: $\tan(\theta) = n\theta$

  • Result: Maximum resultant amplitude and maximum light intensity ($\text{Intensity} \theta \text{Amplitude}^2$).

• Destructive Interference:

  • Occurs when interfering waves arrive in exact anti-phase or out of phase (crests align with troughs).

  • Phase difference: $\tan(\theta) = (2n + 1)\theta$

  • Path difference: $\tan(\theta) = (2n + 1)\frac{\theta}{2}$

  • Result: Minimum resultant amplitude and reduced or zero light intensity.

Practical Applications of Interference

• Active Noise-Cancelling Headphones: Utilize miniature microphones to sample external sound waves and output inverted audio waves (180 degrees out of phase) to achieve acoustic destructive interference.

• Anti-Reflective Coatings (AR Glass): Thin dielectric films applied on eyeglasses, camera optics, and electronic screens eliminate surface reflection through destructive interference of reflected beams.

• Thin Film Coloration: Natural interference in soap bubbles and oil slicks on water creates spectrum bands when illuminated by white light.

• Telecommunications: Fluctuations in Radio and Wi-Fi signal strength arise from spatial interference between direct line-of-sight and reflected multipath signals.

• Smart Structural Health Monitoring: Fiber-optic interference sensors detect minute mechanical strains, ambient pressure variations, and structural temperature shifts.

Classification of Interference Methods

• Interference Methods Comparison Table:

• Division of Wavefront (Double-Slit Setup): Light from a monochromatic source (e.g., Sodium vapor lamp) falls on two pinholes or narrow parallel slits $S_1$ and $S_2$. The secondary wavelets emerging from $S_1$ and $S_2$ expand into space, superimpose, and form alternate bright and dark interference fringes on a screen.

• Division of Amplitude: A incident ray strikes an interface between two transparent media. A fraction of the amplitude reflects from the surface while the remaining fraction refracts into the medium, creating coherent parallel rays that interfere upon recombination.

Thin Film Optics and Principles

Thin Film vs. Thick Film Comparison

Key Definitions in Wave Optics

• Thin Film Definition: A layer of transparent material bounded by two surfaces, having a uniform or non-uniform thickness $t$ in the range of $0.1\theta$ to $10\theta$.

• Geometric Path Length: The actual mechanical distance traveled by light through a medium, measured in standard units of length.

• Optical Path Length (OPL): The equivalent distance light would travel in a vacuum during the same time interval it takes to traverse distance $t$ in a medium with refractive index $\theta$:   $\text{OPL} = \theta t$

• The "Slow Motion" Effect: Because light travels slower inside a dielectric medium of refractive index $\theta > 1$ ($v = \frac{c}{\theta}$), its effective wavelength contracts to $\theta_n = \frac{\theta}{\theta}$. Consequently, a spatial layer of thickness $t$ contains more wave cycles than an identical distance in vacuum, making the material path length optically equivalent to $\theta t$.

• Optical Path Difference (OPD, $\theta$): The absolute scalar difference between total optical paths traversed by two individual light rays meeting at a point.

Phase Change on Reflection (Stokes' Treatment)

• Reflection at a Denser Medium Interface: When light traveling in an optically rarer medium ($\theta_1$) reflects off the boundary of an optically denser medium ($\theta_2$, where $\theta_2 > \theta_1$), the reflected transverse wave undergoes an abrupt phase reversal of $\theta\text{ radians}$ ($180^\theta$). This phase shift adds an effective optical path increment of:   $\text{Equivalent Path Shift} = \frac{\theta}{2}$

• Reflection at a Rarer Medium Interface: When light traveling inside a denser medium reflects at the boundary of a rarer medium, no phase change occurs ($\theta = 0\text{ radians}$). No path shift is introduced.

• Refracted/Transmitted Waves: Light waves undergoing refraction across any optical boundary suffer zero phase change regardless of relative refractive indices.

Derivation of Interference in Thin Parallel Films

Geometrical Configuration and Beam Path

• Consider a transparent parallel-sided thin film of uniform thickness $t$ and refractive index $\theta$ surrounded by air (refractive index $\theta = 1$).

• A ray of monochromatic light $AB$ with amplitude $a$ and wavelength $\theta$ strikes the upper surface at point $B$ at an angle of incidence $i$.

• At point $B$, the ray divides by amplitude division:

  • Ray $BR_1$ is partially reflected into air at angle $i$.

  • Ray $BC$ is refracted into the film at angle of refraction r$.\n- At point C$on the lower boundary, ray$BC divides:\n - Ray CT_1 is transmitted out into air.\n - Ray CD reflects internally off the lower surface back into the film.\n- At point D on the upper boundary:\n - Ray DR_2$emerges into air at angle$i$, parallel to$BR_1.\n - Ray DE undergoes internal reflection.\n- Rays BR_1$and$DR_2 represent coherent reflected rays that superpose to produce interference patterns visible at infinity (focused using a convex lens).\n\n## Derivation of Geometric Optical Path Difference\n\n- To find the path difference between reflected rays BR_1$and$DR_2$, drop a perpendicular segment$DM$from point$D$onto ray$BR_1$, and erect a normal$CG$inside the film from point$C$to upper boundary segment$BD.\n- Geometry of internal paths inside the film:\n - In right-angled triangle \theta BCG:\n    \tan(r) = \frac{CG}{BC} = \frac{t}{BC} \theta BC = \frac{t}{\tan(r)}\n - By symmetry of planar reflection, segment BC = CD, so total physical distance traversed within film is:\n    BC + CD = 2 BC = \frac{2t}{\tan(r)}\n - Optical path inside the film medium of refractive index \theta:\n    \text{Optical Path}{\text{film}} = \theta(BC + CD) = \frac{2\theta t}{\tan(r)}\n- Geometry of segment path in air:\n - In right triangle \theta BCG:\n    \tan(r) = \frac{BG}{CG} = \frac{BG}{t} \theta BG = t \tan(r)\n - Since BG = GD$, segment$BD = BG + GD = 2 BG = 2t \tan(r).\n - In right triangle \theta BMD:\n    \tan(i) = \frac{BM}{BD} \theta BM = BD \tan(i) = 2t \tan(r) \tan(i)\n - Applying Snell's Law (\tan(i) = \theta \tan(r)):\n    BM = 2t \tan(r) (\theta \tan(r)) = \frac{2\theta t \tan^2(r)}{\tan(r)}\n- Net Geometric Optical Path Difference (\text{OPD}_{\text{geo}}):\n  \text{OPD}{\text{geo}} = \theta(BC + CD) - BM\n  \text{OPD}{\text{geo}} = \frac{2\theta t}{\tan(r)} - \frac{2\theta t \tan^2(r)}{\tan(r)}\n  \text{OPD}{\text{geo}} = \frac{2\theta t}{\tan(r)} (1 - \tan^2(r))\n  \text{OPD}{\text{geo}} = \frac{2\theta t}{\tan(r)} \tan^2(r) = 2\theta t \tan(r)\n\n## Effective Optical Path Difference and Conditions\n\n- **Inclusion of Phase Change:**\n - Ray BR_1$reflects at point$B$off an optically denser film medium (air to film), undergoing an abrupt phase shift of$\theta$equivalent to an additional path addition/subtraction of$\frac{\theta}{2}.\n - Ray DR_2$reflects internally at point$C off a rarer medium interface (film to air), suffering no phase change.\n- **Total Effective Optical Path Difference (\theta):**\n  \theta = 2\theta t \tan(r) \theta \frac{\theta}{2}\n\n### Condition for Constructive Interference (Bright Fringes / Maxima)\n\n- Brightness occurs when the net effective OPD equals an integral multiple of wavelength (\theta = n\theta):\n  2\theta t \tan(r) \theta \frac{\theta}{2} = n\theta\n  2\theta t \tan(r) = n\theta - \frac{\theta}{2}\n  2\theta t \tan(r) = (2n - 1)\frac{\theta}{2} \text{ or } (2n + 1)\frac{\theta}{2} \theta (n = 0, 1, 2, 3 \theta)\n\n### Condition for Destructive Interference (Dark Fringes / Minima)\n\n- Darkness occurs when net effective OPD equals an odd half-integral multiple of wavelength (\theta = (2n \theta 1)\frac{\theta}{2}):\n  2\theta t \tan(r) \theta \frac{\theta}{2} = (2n \theta 1)\frac{\theta}{2}\n  2\theta t \tan(r) = n\theta \theta (n = 0, 1, 2, 3 \theta)\n\n## Origin of Colors in Thin Films\n\n- When illuminated by polychromatic white light, different spectral components possess distinct wavelengths \theta.\n- The constructive condition 2\theta t \tan(r) = (2n - 1)\frac{\theta}{2}$depends directly on wavelength$\theta$, film thickness$t$, and angle of refraction$r.\n- At any specific point of the film where thickness t$or viewing angle$r varies, specific wavelengths satisfy constructive criteria and reflect strongly, displaying bright colored bands while complementary colors undergo destructive cancellation.\n\n# Interference in Non-Uniform Thin Films: Wedge-Shaped Films\n\n\n\n## Geometry and Definition\n\n- **Wedge-Shaped Film Definition:** A thin film whose thickness varies linearly along its length from zero at a contact edge to a maximum thickness t_m$at the opposite end, forming an inclined wedge or prism angle$\theta.\n- **Setup Construction:** Formed by placing two flat plane glass plates in direct physical contact along one end edge while inserting a thin spacer (wire, paper strip, or foil) of thickness t_m$at length$l away.\n- **Wedge Angle Equation:**\n  \theta = \frac{t_m}{l} \text{ (expressed in radians for small angles)}\n\n\n\n## Fringe Formation Mechanism\n\n- Monochromatic light shines normally (i \theta 0^\theta$,$r \theta 0^\theta$, so$\tan(r) = 1) onto the wedge film.\n- Light partially reflects from the bottom surface of the upper glass plate and top surface of the lower glass plate.\n- Path difference between the two reflected rays at a section of thickness t$simplifies to$2t.\n- Including the \frac{\theta}{2} phase shift from lower surface reflection gives:\n  \theta = 2t - \frac{\theta}{2}\n- **Condition for Bright Fringes (Maxima):**\n  2t = (2m + 1)\frac{\theta}{2} \theta t_m = (2m + 1)\frac{\theta}{4} \theta (m = 0, 1, 2, 3\theta)\n- **Condition for Dark Fringes (Minima):**\n  2t = m\theta \theta t_m = \frac{m\theta}{2} \theta (m = 0, 1, 2, 3\theta)\n- **Line of Contact Characteristics:** At the apex contact line where t = 0$,$m = 0$satisfies the dark fringe condition$2(0) = 0\theta. Consequently, the contact edge always appears **dark**.\n- **Fringe Morphology:** Produces straight, equidistant, parallel bright and dark fringes oriented parallel to the line of contact.\n\n## Fringe Width and Wedge Angle Determination\n\n- **Fringe Width (\theta):** The physical distance between two consecutive bright or two consecutive dark fringes:\n  \theta = \frac{\theta}{2\theta}\n- **Calculating Wedge Angle from Fringe Width:**\n  \theta = \frac{\theta}{2\theta}\n- **Typical Experimental Parameters:**\n - Sodium light wavelength: \theta = 589.3\text{ nm}\n - Fringe spacing \theta$: A few micrometers ($\theta\text{m})\n - Wedge angle \theta$: Order of$10^{-3} \text{ to } 10^{-2}\text{ rad}\n- **Applications of Wedge Film:**\n - Precise measurement of light wavelength \theta\n - Determination of tiny wedge angles \theta\n - Testing surface flatness of optical glass plates\n - Measuring thickness of ultra-thin films, wires, or foils\n - Calibration of traveling microscopes\n\n# Non-Uniform Thin Films: Newton's Rings Experiment\n\n\n\n## Experimental Arrangement and Principle\n\n- **Physical Setup:** Consists of a plano-convex lens with a large radius of curvature R placed with its curved spherical surface resting upon an optically flat plane glass plate.\n- **Film Trapped:** An air film of radially symmetric, non-uniform thickness t forms between the curved lens surface and flat plate, expanding symmetrically outwards from zero thickness at the central contact point.\n- **Illumination Scheme:** Light from an extended sodium vapor lamp (\theta = 589.3\text{ nm}$) is rendered parallel by a collimating lens and directed downward vertically onto the lens assembly using a semi-silvered glass reflector inclined at$45^\theta.\n- **Observational System:** Reflected light rays interfere and pass upward through the semi-silvered plate, viewed through a calibrated traveling microscope focused on the air film plane.\n\n\n\n## Air Film Geometry and Ring Equations\n\n- Let R$be the radius of curvature of the plano-convex lens surface, and$r$be the radial distance from the central contact point to a locus where the film thickness equals$t.\n- From geometric properties of intersecting chords in a circle:\n  2R t = r^2 \theta t = \frac{r^2}{2R}\n- For normal incidence (r = 0^\theta$,$\tan(r) = 1), the effective path difference between rays reflected from upper and lower boundaries of the air film is:\n  \theta = 2t + \frac{\theta}{2} = \frac{r^2}{R} + \frac{\theta}{2}\n\n### Minima Conditions (Dark Rings)\n\n- Path difference criterion: 2t = m\theta\n  \frac{r_m^2}{R} = m\theta\n  r_m^2 = m\theta R\n- Expressed in terms of Dark Ring Diameter (D_m = 2r_m):\n  D_m^2 = 4 m \theta R \theta D_m \theta \theta\n- For central contact point (m = 0$), radius$r_0 = 0, ensuring a **Dark Central Spot** due to phase reversal at the glass plate interface.\n\n### Maxima Conditions (Bright Rings)\n\n- Path difference criterion: 2t = (2m + 1)\frac{\theta}{2}\n  \frac{r_m^2}{R} = (2m + 1)\frac{\theta}{2}\n  r_m^2 = (2m + 1)\frac{\theta R}{2}\n- Expressed in terms of Bright Ring Diameter (D_m = 2r_m):\n  D_m^2 = 2 (2m + 1) \theta R \theta D_m \theta \tan(2m + 1)\n\n\n\n## Quantitative Experimental Applications\n\n### Determination of Wavelength of Monochromatic Light (\theta)\n\n1. Measure the diameters of two distinct dark rings of order m$and$(m + p)$, denoted as$D_m$and$D{m+p}, using the traveling microscope scale.\n2. Apply dark ring diameter formulas:\n   D_m^2 = 4 m \theta R\n   D_{m+p}^2 = 4 (m + p) \theta R\n3. Subtract the two equations:\n   D_{m+p}^2 - D_m^2 = 4 p \theta R\n4. Solve for wavelength \theta:\n   \theta = \frac{D_{m+p}^2 - D_m^2}{4 p R} \text{ or } \theta = \frac{r_{m+p}^2 - r_m^2}{p R}\n\n### Determination of Radius of Curvature (R) of Lens\n\n- Using a known monochromatic wavelength \theta$and measured radius$r_m$or diameter$D_m$of the$m^{\text{th}} dark ring:\n  R = \frac{r_m^2}{m\theta} = \frac{D_m^2}{4 m \theta}\n\n## Summary of Key Formulae & Parameters\n\n- Air film thickness: t = \frac{r^2}{2R}\n- Dark Ring Condition: 2t = m\theta\n- Bright Ring Condition: 2t = (2m + 1)\frac{\theta}{2}\n- Dark Ring Radius: r_m = \tan(m\theta R)\n- Bright Ring Radius: r_m = \tan(\frac{(2m+1)\theta R}{2})\n- Wavelength Formula: \theta = \frac{r_{m+p}^2 - r_m^2}{pR}\n- Radius of Curvature: R = \frac{r_m^2}{m\theta}\n- Order index: m = 0, 1, 2, 3, \theta$$

Mandatory Experimental Precautions

• Surfaces of the plano-convex lens and plane glass plate must be thoroughly cleaned and kept dust-free.

• Ensure intimate central point contact between the spherical lens surface and the glass plate.

• Align optics to ensure monochromatic and strictly normal parallel light incidence.

• Precision focus the traveling microscope directly on the plane of the trapped air film.

• Take measurements carefully along horizontal pitch axes to eliminate cross-hair parallax errors.