Wave Optics PDF

Wave optics 1. Huygens' Principle and Wavefronts Huygens' Principle: Every point on a wavefront acts as a source of secondary spherical wavelets that spread out in all directions. The new wavefront is the tangential surface to these secondary wavelets. Wavefront Types:...

Wave optics 1. Huygens' Principle and Wavefronts Huygens' Principle: Every point on a wavefront acts as a source of secondary spherical wavelets that spread out in all directions. The new wavefront is the tangential surface to these secondary wavelets. Wavefront Types: o Plane Wavefront: Produced by sources at an infinite distance. o Spherical Wavefront: Produced by a point source. o Cylindrical Wavefront: Produced by a linear source. Applications: Huygens' principle explains reflection, refraction, and diffraction. 2. Interference of Light Principle of Superposition: When two light waves overlap, their amplitudes add up. Types of Interference: o Constructive Interference: When crest meets crest (bright fringe), path difference = nλn\lambdanλ. o Destructive Interference: When crest meets trough (dark fringe), path difference = (n+1/2)λ(n + 1/2) \lambda(n+1/2)λ. Young’s Double-Slit Experiment (YDSE): o Fringe Width (β)( \beta )(β): The distance between two consecutive bright or dark fringes. β=λDd\beta = \frac{\lambda D}{d}β=dλD where DDD is the distance to the screen, ddd is slit separation, and λ\lambdaλ is the wavelength. Coherence: o Temporal Coherence: Consistency of phase over time. o Spatial Coherence: Consistency of phase across the wavefront. 3. Thin Film Interference Interference in Thin Films: Occurs due to the reflection of light from the upper and lower surfaces of a thin film. Conditions for Interference:  o Constructive: 2μtcos⁡r=nλ2 \mu t \cos r = n \lambda2μtcosr=nλ (where μ\muμ is refractive index, ttt is thickness, rrr is angle of refraction, and nnn is an integer). o Destructive: 2μtcos⁡r=(2n+1)λ22 \mu t \cos r = (2n+1) \frac{\lambda}{2}2μtcosr=(2n+1)2λ. Applications: Colors in soap bubbles, anti-reflective coatings. 4. Diffraction of Light Types of Diffraction: o Fresnel Diffraction: Source or screen is close to the slit. o Fraunhofer Diffraction: Source and screen are at infinite distances, producing parallel rays. Single-Slit Diffraction: o Central Maximum: Brightest fringe, width 2θ2 \theta2θ (where θ≈λa\theta \approx \frac{\lambda}{a}θ≈aλ , aaa is slit width). o Intensity Pattern: Maximum at the center, decreasing towards edges. Diffraction Grating: o Grating Equation: dsin⁡θ=nλd \sin \theta = n \lambdadsinθ=nλ, where ddd is grating spacing. o Applications: Spectroscopy and wavelength measurement. 5. Polarization of Light Unpolarized Light: Has vibrations in all directions perpendicular to the direction of propagation. Polarization Methods: o Reflection: Light gets partially polarized upon reflection. o Refraction: Birefringent materials (like calcite) split light into two polarized rays. o Scattering: Polarization occurs when light scatters in the atmosphere. Malus' Law: I=I0cos⁡2θI = I_0 \cos^2 \thetaI=I0 cos2θ, where I0I_0I0 is initial intensity, and θ\thetaθ is the angle between the light’s polarization direction and the analyzer. Brewster’s Law: tan⁡θB=n\tan \theta_B = ntanθB =n (where θB\theta_BθB is Brewster’s angle and nnn is the refractive index of the material). 6. Resolving Power of Optical Instruments Rayleigh Criterion: Two sources are resolved if the first minimum of one diffraction pattern coincides with the maximum of the other. Microscope Resolving Power: R=2nsin⁡θλR = \frac{2n \sin \theta}{\lambda}R=λ2nsinθ , where nnn is the refractive index, θ\thetaθ is the angle of light collection, and λ\lambdaλ is the wavelength. Telescope Resolving Power: R=D1.22λR = \frac{D}{1.22 \lambda}R=1.22λD , where DDD is the diameter of the objective lens. Applications: Used in designing lenses and optical instruments with higher precision. o. Tips for Exam Preparation: 1. Understand Core Principles: Focus on understanding Huygens' principle, interference patterns, and diffraction concepts thoroughly. 2. Practice Problems: Go through numerical problems related to Young’s Double- Slit Experiment, thin-film interference, and diffraction calculations. 3. Memorize Key Formulas: Make sure to memorize the formulas for interference and diffraction, as well as laws related to polarization and resolving power. 4. Visualize with Diagrams: Draw neat diagrams for each concept to reinforce understanding and illustrate answers. 5. Revisit Past Papers: Review past exam questions on these topics to get familiar with question patterns. derivation Using Huygens' Principle to Derive the Laws of Reflection and Refraction a) Law of Reflection 1. Define the Incident Wavefront: a. Imagine a plane wavefront ABABAB hitting a reflecting surface XYXYXY at an angle of incidence iii. b. Each point on ABABAB acts as a source of secondary wavelets that spread in all directions.  2. Apply Huygens' Principle: a. From point BBB on the wavefront, draw a secondary wavelet moving at speed vvv. b. After a time ttt, the wavelet will have traveled a distance BC=vtBC = vtBC=vt. 3. Draw the Reflected Wavefront: a. The new wavefront (reflected wavefront) will touch the tangents from all secondary wavelets. b. Using geometry, show that the angle of incidence iii equals the angle of reflection rrr. b) Law of Refraction (Snell’s Law) 1. Define the Refracted Wavefront: a. A plane wavefront ABABAB moves from medium 1 (with speed v1v_1v1 ) to medium 2 (with speed v2v_2v2 ), forming a refracted wavefront CDCDCD. 2. Use Geometry: a. Derive that sin⁡i/sin⁡r=v1/v2\sin i / \sin r = v_1 / v_2sini/sinr=v1 /v2 , and by introducing refractive indices n1n_1n1 and n2n_2n2 , you get Snell's law: n1sin⁡i=n2sin⁡rn_1 \sin i = n_2 \sin rn1 sini=n2 sinr. 2. Young's Double-Slit Experiment (YDSE) 1. Set Up the Experiment: a. Two slits separated by distance ddd are illuminated by light of wavelength λ\lambdaλ. b. The screen is placed at distance DDD. 2. Calculate Path Difference: a. At a point PPP on the screen at distance yyy from the central line, the path difference is Δx=dsin⁡θ≈dyD\Delta x = d \sin \theta \approx \frac{d y}{D}Δx=dsinθ≈Ddy. 3. Interference Condition: a. Constructive Interference (bright fringe) happens when Δx=nλ\Delta x = n \lambdaΔx=nλ. b. Destructive Interference (dark fringe) occurs when Δx=(n+12)λ\Delta x = (n + \frac{1}{2}) \lambdaΔx=(n+21 )λ. 4. Fringe Width β\betaβ: a. Derive β=λDd\beta = \frac{\lambda D}{d}β=dλD for the distance between bright or dark fringes. 3. Interference in Thin Films 1. Incident Light on Thin Film: a. Light reflects from the top and bottom surfaces of a thin film of thickness ttt and refractive index μ\muμ. 2. Path Difference Calculation: a. For the two reflected rays, the path difference is 2μtcos⁡r2 \mu t \cos r2μtcosr. 3. Interference Conditions: a. Constructive: 2μtcos⁡r=mλ2 \mu t \cos r = m \lambda2μtcosr=mλ. b. Destructive: 2μtcos⁡r=(m+12)λ2 \mu t \cos r = (m + \frac{1}{2}) \lambda2μtcosr=(m+21 )λ. 4. Single-Slit Diffraction 1. Single-Slit Setup: a. Light of wavelength λ\lambdaλ passes through a slit of width aaa. 2. Condition for Minima: a. At an angle θ\thetaθ, if asin⁡θ=mλa \sin \theta = m \lambdaasinθ=mλ, a minimum is formed (for m=1,2,…m = 1, 2, \ldotsm=1,2,…). 3. Central Maximum Width: a. The width of the central maximum is approximately 2θ≈2λa2 \theta \approx \frac{2 \lambda}{a}2θ≈a2λ. 5. Diffraction Grating Equation 1. Multiple Slit Setup: a. Light passes through a grating with slit spacing ddd. 2. Derive Grating Equation: a. Constructive interference occurs at dsin⁡θ=nλd \sin \theta = n \lambdadsinθ=nλ, where n=0,1,2,…n = 0, 1, 2, \ldotsn=0,1,2,…. 6. Malus' Law in Polarization 1. Intensity of Polarized Light: a. When light passes through an analyzer at an angle θ\thetaθ, the intensity I=I0cos⁡2θI = I_0 \cos^2 \thetaI=I0 cos2θ. 7. Resolving Power of Optical Instruments a) Microscope Resolving Power R=2nsin⁡θλR = \frac{2 n \sin \theta}{\lambda}R=λ2nsinθ. b) Telescope Resolving Power R=D1.22λR = \frac{D}{1.22 \lambda}R=1.22λD.