Unit 06: Optics • Chapter 10 (Wave Optics)
Wavefronts, Huygens' Principle & Wave Proofs
Surface of constant phase, secondary wavelet envelope construction, wave-theory derivation of the Laws of Reflection & Refraction (Snell's Law), and optical element wavefront shaping.
Socho ek shant jheel (calm pond) mein patthar girane par kya hota hai: girne wale point se chaaron taraf concentric circular ripples (waves) bahar ki taraf failti hain. Har circle par mojood paani ke saare boondein ek sath upar (crest) aur ek sath neeche (trough) jaati hain. Yahi continuous line ya surface jisme har particle ka phase bilkul same ho, Wavefront kehlata hai! Christiaan Huygens ne 1678 mein ek krantikari baat samjhayi: wave aage kaise badhti hai? Wavefront par baitha har ek point khud ek naya mini-source (secondary source) ban jata hai jo chaaron taraf spherical wavelets phenkta hai. Ek second baad in saare wavelets par aage se khenchi gayi common tangent (forward envelope) agla naya wavefront banati hai! Light ray kuch aur nahi, balki is wavefront ke surface par kheench gaya ek perpendicular arrow (normal) hai jo energy propagation ki direction batata hai!
1. Geometric Shapes of Wavefronts
Depends strictly on the geometry of the primary source:
- Spherical Wavefront: Generated by a point source ($S$) at finite distance. Amplitude $A \propto \frac{1}{r}$, Intensity $I \propto \frac{1}{r^2}$.
- Cylindrical Wavefront: Generated by a linear slit source. Amplitude $A \propto \frac{1}{\sqrt{r}}$, Intensity $I \propto \frac{1}{r}$.
- Plane Wavefront: Produced by any source (point or line) located at infinite distance ($\infty$), e.g., sunlight reaching Earth. Amplitude and intensity remain constant along propagation.
2. Huygens' Two Postulates
Kinematic mechanism of wave propagation:
- Every point on a given primary wavefront acts as a fresh source of new disturbance, emitting spherical secondary wavelets spreading out in all directions with the speed of light in that medium ($v$).
- The forward envelope (tangential surface touching all secondary wavelets in the forward direction) at time $t + \Delta t$ gives the position of the new secondary wavefront. Backward wavelets carry zero amplitude due to inclination factor $\frac{1 + \cos\theta}{2}$.
Proof of the Law of Reflection ($\angle i = \angle r$) Using Huygens' Principle
Fig 10.1A: Reflection of a plane wavefront $AB$ incident at angle $i$ on reflecting surface $XY$, forming reflected wavefront $CD$ at angle $r$.
1. Consider a plane wavefront $AB$ incident obliquely on a plane reflecting surface $XY$ at an angle $i$. At time $t = 0$, edge $A$ touches the mirror surface, while edge $B$ has to travel a distance $BC$ to reach point $C$ on the mirror.
2. If $v$ is the speed of light in the medium, the time taken by point $B$ to reach $C$ is $\tau = \frac{BC}{v} \implies BC = v \tau$.
3. According to Huygens' principle, the moment point $A$ touches the surface, it becomes a source of secondary wavelets radiating hemispherical wavelets of radius $v \tau$ into the medium. At time $\tau$, let a tangent $CD$ be drawn from point $C$ to this wavelet sphere. Then $AD = v \tau$.
4. The tangent plane $CD$ represents the Reflected Wavefront.
5. In right-angled triangles $\triangle ABC$ and $\triangle ADC$:
• $\angle ABC = \angle ADC = 90^\circ$ (Rays are perpendicular to wavefronts)
• $BC = AD = v \tau$ (Distances traveled in identical time $\tau$ with identical speed $v$)
• $AC = AC$ (Common hypotenuse)
6. By the RHS Congruence Criterion: $\triangle ABC \cong \triangle ADC$. Therefore, corresponding angles must be equal:
Since the incident ray, the reflected ray, and the normal to the reflecting surface at point $A$ all lie in the plane of the paper, both classical laws of reflection are proved rigorously.
Proof of Snell's Law of Refraction ($\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{\mu_2}{\mu_1}$) Using Huygens' Principle
Fig 10.1B: Refraction of plane wavefront $AB$ incident at angle $i$ from medium 1 ($v_1$) into medium 2 ($v_2 < v_1$), forming refracted wavefront $CD$ at angle $r$.
1. Let a plane wavefront $AB$ propagate from medium 1 (speed $v_1$, index $\mu_1$) towards medium 2 (speed $v_2$, index $\mu_2$, where $\mu_2 > \mu_1 \implies v_2 < v_1$), incident at angle $i$ on boundary interface $AC$.
2. Let $\tau$ be the time taken by point $B$ of the wavefront to travel distance $BC$ in medium 1: $BC = v_1 \tau$.
3. During this same time interval $\tau$, the secondary spherical wavelet originating from point $A$ spreads out into medium 2 over a smaller radius: $AD = v_2 \tau$.
4. The tangent plane $CD$ drawn from $C$ to this wavelet sphere represents the Refracted Wavefront. Since rays are perpendicular to wavefronts, $\angle ABC = 90^\circ$ and $\angle ADC = 90^\circ$.
5. In right-angled triangle $\triangle ABC$:
6. In right-angled triangle $\triangle ADC$:
7. Dividing the two trigonometric equations:
8. By definition of absolute refractive index, $\mu_1 = \frac{c}{v_1}$ and $\mu_2 = \frac{c}{v_2}$, which gives $\frac{v_1}{v_2} = \frac{c/\mu_1}{c/\mu_2} = \frac{\mu_2}{\mu_1}$:
Wave Theory Victory: Since $\mu_2 > \mu_1 \implies \frac{\sin i}{\sin r} > 1 \implies \sin i > \sin r \implies i > r$. This mathematically proves that when light enters a denser medium, it must bend towards the normal and slow down ($v_2 < v_1$), debunking Newton's corpuscular theory which incorrectly predicted light accelerates in denser media!
Behavior of a Plane Wavefront Passing Through Optical Components
1. Through a Thin Prism
The lower portion of the incident plane wavefront travels through maximum thickness of glass (slower speed) while the upper portion travels through air. Hence, the emergent wavefront tilts towards the base, emerging as an inclined plane wavefront.
2. Through a Convex Lens
The central ray passes through the thickest part of the glass and suffers maximum time delay. The outer edges travel mostly through air. Hence, a plane wavefront emerges as a converging spherical wavefront collapsing onto the real focus $F$.
3. Off a Concave Mirror
The outer edges strike the mirror first and reflect back sooner than the central portion which travels deeper to the pole. Hence, the reflected wavefront emerges as a converging spherical wavefront centered at focus $F$.
Trap 1: Frequency and Phase Constancy Across Media
*Board Question:* When monochromatic light of wavelength $\lambda$ travels from air into a medium of refractive index $\mu$, what happens to its frequency, speed, and wavelength?
• Frequency $\nu$ remains STRICTLY CONSTANT! Frequency is determined solely by the oscillating source charges, not by the medium.
• Speed drops: $v = \frac{c}{\mu}$.
• Wavelength decreases: $\lambda' = \frac{v}{\nu} = \frac{c/\mu}{\nu} = \mathbf{\frac{\lambda}{\mu}}$.
Trap 2: Why Backward Wavefronts Do Not Exist
*Exam Trap:* If every point on a wavefront emits spherical secondary wavelets in all directions, why does a wave not travel backwards?
Huygens could not explain this mathematically. Later, Voigt and Kirchhoff proved that the amplitude of a secondary wavelet in direction $\theta$ with forward normal is governed by the Inclination Factor (Obliquity Factor):
$$F(\theta) = \frac{1 + \cos\theta}{2}$$
• Forward direction ($\theta = 0^\circ$): $F(0) = \frac{1 + 1}{2} = \mathbf{1}$ (Maximum).
• Backward direction ($\theta = 180^\circ$): $F(\pi) = \frac{1 - 1}{2} = \mathbf{0}$ (Zero backward propagation).
Trap 3: Phase Difference Between Points on the Same Wavefront
*Direct Conceptual Trap:* What is the phase difference between any two arbitrary points lying on the exact same wavefront?
*Correct Answer:* ZERO ($\Delta\phi = 0$)!
By definition, a wavefront is the locus of all oscillating medium points having the identical phase of vibration at that instant.
Trap 4: Corpuscular vs Wave Theory on Light Speed in Water
Newton's Corpuscular theory assumed that light particles are attracted by denser medium particles, predicting $v_{\text{water}} > v_{\text{air}}$. Huygens' wave theory predicted $v_{\text{water}} < v_{\text{air}}$. In 1850, Foucault's direct experiment measured the speed of light in water and confirmed $v_{\text{water}} < v_{\text{air}}$, decisively proving the wave theory of light!
Problem: Light of wavelength $589\text{ nm}$ is incident from air on a water surface. (a) What are the wavelength, frequency, and speed of reflected light? (b) What are the wavelength, frequency, and speed of refracted light? (Take $\mu_{\text{water}} = 1.33$).
Solution: In air: $\lambda = 589\text{ nm} = 5.89 \times 10^{-7}\text{ m}$, $c = 3.0 \times 10^8\text{ m/s}$.
Frequency in air (and in all media):
(a) For Reflected Light (Traveling back in Air):
• Speed: $v = c = 3.0 \times 10^8\text{ m/s}$.
• Frequency: $\nu = 5.09 \times 10^{14}\text{ Hz}$.
• Wavelength: $\lambda = 589\text{ nm}$.
(b) For Refracted Light (Traveling in Water, $\mu = 1.33$):
Problem: A plane wavefront is incident on a thin glass prism of refracting angle $4^\circ$ and refractive index $1.5$. Calculate the angle by which the emergent wavefront is tilted relative to the incident wavefront. If this same wavefront is passed through a convex lens of focal length $20\text{ cm}$, what will be the shape and radius of curvature of the emergent wavefront after traveling $10\text{ cm}$ beyond the lens?
Solution:
1. For a thin prism ($A = 4^\circ \le 10^\circ$), the deviation produced in every ray is:
Since rays remain perpendicular to the wavefront, the emergent plane wavefront is tilted by exactly $2.0^\circ$ towards the base.
2. For a convex lens of $f = +20\text{ cm}$, an incident plane wavefront ($u = \infty$) is transformed into a converging spherical wavefront focused towards $F = 20\text{ cm}$.
At the lens aperture, the emergent wavefront has radius of curvature $R = f = 20\text{ cm}$. After traveling distance $d = 10\text{ cm}$, the wavefront has converged closer to the focus:
Wavefront Phase
$\Delta\phi = 0$ (Constant Phase)
Snell's Law Proof
$\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{\mu_2}{\mu_1}$
Frequency Invariance
$\nu = \text{const}, \lambda' = \frac{\lambda}{\mu}$
Ray vs Wavefront
$\vec{\text{Ray}} \perp \text{Wavefront}$