Wave Optics NCERT

WAVE OPTICS (Chapter 10)

10.1 INTRODUCTION

• History of Light Theories:

  • 1637: Descartes proposed the corpuscular model of light which derived Snell’s Law, explaining reflection and refraction.

  • Predicted that if a ray of light bends towards the normal during refraction, the speed in the second medium would be greater.

  • Isaac Newton further developed this model in OPTICKS. This model became widely attributed to him due to the book's popularity.

  • 1678: Christiaan Huygens proposed the wave theory of light.

  • The wave model explained refraction differently: if the wave bends towards the normal, the speed of light is less in the second medium.

  • Experiments confirmed this prediction, especially Foucault’s experiment in 1850, which showed light travels slower in water than in air.

  • Challenges: Wave theory was slow to gain acceptance due to:

  • Newton’s authority and the belief that waves require a medium to propagate.

  • The propagation of light through a vacuum seemed contradictory to wave concepts.

  • Maxwell's Contribution:

  • Developed electromagnetic theory of light, describing light as electromagnetic waves.

  • Derived wave equations predicting electromagnetic waves, leading to the realization that light can propagate through vacuum.

  • Light waves consist of changing electric and magnetic fields.

10.2 HUYGENS PRINCIPLE

• Wavefront Definition:

  • A wavefront is described as a surface of constant phase. Examples include:

  • Circular waves in a water pool when a stone is dropped.

  • Spherical waves from a point source.

  • Propagation Description:

  • The speed with which the wavefront moves is called the speed of the wave.

  • Energy propagation is perpendicular to the wavefront direction.

  • Huygens Principle: States that:

  • Each point on a wavefront acts as a source of secondary wavelets spreading out in all directions.

  • The envelope of these wavelets determines the new position of the wavefront over time.

  • Example of wavefront propagation illustrated with spherical wave at time t=0 and evolving to time t=Δt.

  • Shortcoming: Huygens’ assumption on wavelet amplitude leading to no backwave is an ad-hoc explanation.

10.3 REFRACTION AND REFLECTION OF PLANE WAVES USING HUYGENS PRINCIPLE

• 10.3.1 Refraction of a Plane Wave:

  • Surface dividing two media denoted as PP′ with speeds v1 and v2.

  • Plane wavefront AB at angle i hits the interface; time taken to travel BC is Δt such that BC = v1Δt.

  • To determine the refracted wavefront shape, construct a sphere of radius v2Δt from point A.

  • Tanner plane CE derives the refracted wavefront.

  • From triangles ABC and AEC:

  • $ext{sin } i = rac{BC}{AC} = rac{v_1 au}{AC}$

  • $ext{sin } r = rac{AE}{AC} = rac{v_2 au}{AC}$

  • Results in the derivation: $rac{v_1}{v_2} = rac{ ext{sin } i}{ ext{sin } r}$ (Snell’s Law).

  • This implies when $r < i$, v_2 < v_1 confirming wave theory's values.

  • Refractive Index:

  • Defined as $n = rac{c}{v}$; thus, $n_1 ext{sin } i = n_2 ext{sin } r$.

  • The wavelength in each medium is related by $rac{ ext{λ}_1}{ ext{λ}_2} = rac{v_1}{v_2}$.

• 10.3.2 Refraction at a Rarer Medium:

  • Here, v_2 > v_1 leading to angle of refraction greater than the angle of incidence.

  • Introduction of critical angle $i_c$ where, $ext{sin } i_c = rac{n_2}{n_1}$ results in total internal reflection when i > i_c.

• 10.3.3 Reflection of Plane Waves:

  • Light reflects from surface MN at angle i.

  • Wavefronts are constructed similarly to refraction demonstrating that the angle of incidence equals the angle of reflection, thus confirming the law of reflection.

10.4 COHERENT AND INCOHERENT ADDITION OF WAVES

• Interference Patterns:

  • Based on the principle of superposition: resultant displacement is the vector sum of displacements from individual waves.

  • Two coherent sources can produce constructive (maximum intensity) or destructive (minimum intensity) interference, dictated by path differences.

  • Examples:

  • Constructive interference when path differences yield multiples of the wavelength (e.g., 0, 1, 2…

  • Destructive at half-integral multiples.

10.5 INTERFERENCE OF LIGHT WAVES AND YOUNG’S EXPERIMENT

• Young's Experiment:

  • Parallel sources S1 and S2 producing coherent light waves create an interference pattern.

  • Bright fringes appear where path difference equals integral multiples of the wavelength, and dark fringes between these points.

10.6 DIFFRACTION

• General Characteristics: All waves exhibit diffraction, leading to alternate dark and bright regions near geometrical shadows.

  • Explained through single slit experiments demonstrating spreading of light.

  • Diffraction effects are pronounced when slits or obstacles are of comparable size to the wavelength of light.

10.7 POLARISATION

• Polarisation Phenomenon: Light waves are transverse, having electric fields at right angles to wave propagation.

  • Introduction to polarising filters that align electric fields in specific orientations.

  • Malus' Law: Describes intensity after light passes through two polaroids as $I = I_0 ext{cos}^2 heta$.

  • Applications include sunglasses and cameras, allowing for control over light intensity.

POINTS TO PONDER

1. Waves from a point source diffuse in directions, yet light travels via rays.

2. Interference phenomena elucidate wave behaviors through constructive and destructive interference.

3. Diffraction illustrates light's limits in optical resolution.

4. Polarisation effects are significant primarily for transverse waves.

SUMMARY

1. Huygens Principle states wavefronts consist of secondary waves; leads to laws of reflection/refraction.

2. Superposition principle governs the total light intensity based on combined wave contributions.

3. Young's double-slit generates equidistant bright/dark interference fringes, while single-slit diffraction patterns show central bright maxima.

4. Natural light is unpolarised; polaroids ensure controlled light polarization based on their alignment.