Optics revision notes

Key Concepts & Definitions

Nature has endowed the human eye with the sensitivity to detect electromagnetic waves within a small range (wavelength of about 400 nm to 750 nm), which is called light. The speed of light in vacuum is a fundamental constant, $c = 2.99792458 \times 10^8 \text{ m s}^{-1}$, which is the highest speed attainable in nature. Because the wavelength of light is very small compared to ordinary objects, it can be approximated to travel in straight lines.

Ray & Beam

A ray is the path of light, and a bundle of such rays constitutes a beam of light. In the limit of wavelength tending to zero, a ray is defined as the path of energy propagation.

Corpuscular Theory vs Wave Theory

Descartes and Newton's corpuscular model predicted that light bending towards the normal upon refraction would have a greater speed in the second medium. Huygens' wave theory predicted the opposite. Foucault's experiment in 1850 proved that light is slower in water than in air, confirming the wave theory. Later, Maxwell's EM theory proved light waves can propagate in a vacuum.

Wavefront

The locus of points which oscillate in phase is called a wavefront, defined as a surface of constant phase. The energy travels perpendicular to the wavefront. Wavefronts can be spherical (from a point source) or plane (at large distances from the source).

Coherence

Two sources are coherent if the phase difference between the displacements produced by each wave does not change with time. Independent light sources like sodium lamps undergo abrupt phase changes in 10−1010^{-10}10−10 seconds, making them completely incoherent.

Reflection of Light and Spherical Mirrors

The laws of reflection state that the angle of incidence equals the angle of reflection ($\angle i = \angle r$), and the incident ray, reflected ray, and normal lie in the same plane. These laws are valid at each point on any reflecting surface, plane or curved. The geometric center of a spherical mirror is the pole, and the line joining the pole and the center of curvature is the principal axis.

• Focal Length: For paraxial rays (rays close to the pole making small angles with the principal axis), the reflected rays converge at (or appear to diverge from) the principal focus $F$. The focal length $f$ is given by $f = R/2$, where $R$ is the radius of curvature.
• Mirror Equation: The relation between object distance ($u$), image distance ($v$), and focal length ($f$) is $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$.
• Linear Magnification: The ratio of the height of the image ($h'$) to the object ($h$) is $m = \frac{h'}{h} = -\frac{v}{u}$.
• → [JEE TIP] Trap 1 - Image Velocity: If an object moves at a constant speed towards a mirror, its image does not move at a constant speed. The average speed of the image increases substantially as the object gets closer to the mirror. Longitudinally, $v_{image} = -m^2 v_{object}$. For transverse motion, $v_{image} = m \cdot v_{object}$.

Refraction at Plane Surfaces and Total Internal Reflection

When a beam of light encounters a transparent medium, the direction of propagation of the obliquely incident ray changes, a phenomenon called refraction.

• Snell's Law: $\frac{\sin i}{\sin r} = n_{21} = \frac{n_2}{n_1} = \frac{v_1}{v_2}$, where $n_{21}$ is the refractive index of medium 2 with respect to medium 1.
• → [JEE TIP] Trap 2 - Variable Refractive Index: If the refractive index varies continuously with the y-coordinate $n(y)$, apply Snell's Law in the generalized form: $n(y) \sin\theta = \text{constant}$ at any point along the ray trajectory.
• Lateral Shift and Apparent Depth: Refraction through a rectangular glass slab produces no net deviation, only a lateral shift. When viewing near the normal, the apparent depth $h_1$ is related to real depth $h_2$ by $h_1 = h_2 / n$.
• Total Internal Reflection (TIR): When light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle $i_c$, no refraction occurs and the light is totally reflected. The critical angle is $\sin i_c = n_{21}$ (where medium 1 is denser and 2 is rarer).
• Applications of TIR: Prisms can bend light by $90^\circ$ or $180^\circ$ without losing intensity because the critical angle of crown/flint glass is less than $45^\circ$. Optical fibres use high-quality glass/quartz consisting of a core and cladding (core refractive index > cladding refractive index) to transmit signals over long distances via continuous TIR. Even if the fibre is bent, light travels easily. Silica glass fibres can transmit $>95\%$ of light over a 1 km length due to high purification.

Refraction at Spherical Surfaces and Lenses

For refraction at a single spherical surface of radius $R$ separating media $n_1$ and $n_2$: $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$.

• Lens Maker's Formula: By applying the single surface formula successively to both surfaces of a thin lens, we get $\frac{1}{f} = (n_{21} - 1)(\frac{1}{R_1} - \frac{1}{R_2})$.
• Thin Lens Formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$. The linear magnification is $m = \frac{v}{u}$.
• Power of a Lens: The measure of convergence or divergence is $P = \frac{1}{f}$. SI unit is Dioptre (D) when $f$ is in meters ($1\text{ D} = 1\text{ m}^{-1}$). For a combination of thin lenses in contact, effective power is $P = P_1 + P_2 + ...$ and effective focal length is $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + ...$. Total magnification is $m = m_1 m_2 ...$.
• → [JEE TIP] Trap 3 - Silvering of Lenses: If one surface of a lens is silvered, it acts as a mirror. The equivalent power is $P_{eq} = 2P_L + P_M$. Treat it as a multi-element system (e.g., $P_{eq} = P_{glass} + P_{liquid} + P_{mirror} + \dots$). The system behaves like a mirror of focal length $F = -1/P_{eq}$.
• → [JEE TIP] Displacement Method: To find the focal length of a convex lens using two pins fixed at distance $D$, the lens forms a sharp image at two positions separated by $x$. The focal length is $f = \frac{D^2 - x^2}{4D}$. This requires $D \ge 4f$.
• → [JEE TIP] Cutting of Lenses: If a lens of focal length $f$ is cut longitudinally (along the principal axis), the focal length of each half remains $f$, but intensity halves. If cut transversely (perpendicular to axis), the focal length of each half becomes $2f$.

Refraction Through a Prism

For a triangular prism with refracting angle $A$, the angles of refraction at the two interfaces satisfy $r_1 + r_2 = A$. The total angle of deviation is $\delta = i + e - A$.

• Minimum Deviation ($D_m$): Occurs when the refracted ray inside the prism is parallel to its base, which means $i = e$ and $r_1 = r_2 = A/2$. The refractive index is $n_{21} = \frac{\sin((A+D_m)/2)}{\sin(A/2)}$.
• Thin Prism: For small angle prisms, $D_m \approx (n_{21}-1)A$, meaning they do not deviate light much.
• → [JEE TIP] Dispersion & Deviation Combinations: For a combination of two prisms (crown and flint), we can achieve Dispersion without Deviation (net deviation for mean ray is zero: $\delta_1 + \delta_2 = 0$) or Deviation without Dispersion (net angular dispersion is zero: $\theta_1 + \theta_2 = 0$).

Optical Instruments

• Simple Microscope: A converging lens used to view near objects. Magnification is maximum when the image is at the near point $D = 25\text{ cm}$, giving $m = 1 + \frac{D}{f}$. For relaxed viewing (image at infinity), angular magnification is $m = \frac{D}{f}$.
• Compound Microscope: Uses an objective lens (small $f_o$) and an eyepiece (small $f_e$). The objective forms a real, magnified image which acts as the object for the eyepiece. Total magnification for final image at infinity is $m = \frac{L}{f_o} \times \frac{D}{f_e}$, where $L$ is the tube length.
• Telescope: Used for distant objects. The objective has a large focal length and large aperture to gather light. Magnifying power is $m = \frac{f_o}{f_e}$ and the tube length is $f_o + f_e$ for normal adjustment.
• Reflecting Telescope (Cassegrain): To avoid chromatic aberration and mechanical difficulties of supporting large lenses, modern telescopes use concave parabolic mirrors for the objective. A secondary convex mirror deflects light through a hole in the primary mirror.
• → [JEE TIP] Resolving Power Limits: The main consideration of an astronomical telescope is light gathering power and resolving power. The limit to distinguish close objects is strictly set by the phenomenon of diffraction.

Wave Theory of Light and Huygens Principle

• Huygens Principle: Each point of a wavefront acts as a source of secondary wavelets spreading out with the speed of the wave. The new wavefront at a later time is the forward envelope (common tangent) to these secondary wavelets.
• Backwave Anomaly: Huygens' geometrical construction technically yields a forward wave and a backward wave. Huygens made an adhoc assumption that the amplitude of secondary wavelets is zero in the backward direction to ignore it.
• Refraction Proof: When a plane wavefront strikes a denser medium, the speed decreases ($v_2 < v_1$). Constructing secondary wavelets shows $\frac{\sin i}{v_1} = \frac{\sin r}{v_2}$. Thus $\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$. The wavelength decreases ($\lambda_2 = \lambda_1 \frac{v_2}{v_1}$) but frequency $\nu$ remains strictly constant.

Interference of Light Waves and YDSE

Based on the superposition principle, the resultant displacement is the vector sum of individual displacements.

• Coherent Addition: For two coherent sources generating waves of same amplitude $a$, the resultant intensity is $I = 4I_0 \cos^2(\frac{\phi}{2})$, where $I_0$ is the intensity of an individual source.
• Constructive & Destructive Interference: Constructive interference (bright regions) occurs when path difference $\Delta x = n\lambda$. Destructive interference (dark regions) occurs when $\Delta x = (n+1/2)\lambda$.
• Young's Double Slit Experiment (YDSE): Thomas Young "locked" the phases of two pinholes by illuminating them from a single primary source.
  • Positions of Bright Fringes: $x_n = \frac{n\lambda D}{d}$ ($n = 0, \pm1, \dots$).
  • Positions of Dark Fringes: $x_n = (n + 1/2) \frac{\lambda D}{d}$ ($n = 0, \pm1, \dots$).
• → [JEE TIP] Trap 4 - Insertion of Glass Slab: If a transparent slab of thickness $t$ and refractive index $\mu$ is placed in front of one slit, the entire fringe pattern shifts towards that slit by a distance $\Delta x_{shift} = \frac{D}{d}(\mu - 1)t$. The fringe width $\beta$ remains completely unchanged.
• → [JEE TIP] Trap 5 - Interference in Water: If the entire YDSE apparatus is submerged in a liquid of refractive index $\mu$, the wavelength changes to $\lambda' = \lambda/\mu$. Consequently, the fringe width strictly decreases to $\beta' = \beta/\mu$.
• → [JEE TIP] Shape of Fringes: The locus of a point having a constant path difference from two point sources is a hyperbola in 3D. Hence, the interference fringes on a flat screen are locally hyperbolic, but for $D \gg d$, they appear straight.
• → [JEE TIP] Energy Conservation: In interference and diffraction, light energy is only redistributed. If it reduces in one region (dark fringe), it increases in another (bright fringe). No energy is ever destroyed.

Diffraction of Light

Diffraction is the spreading of light around the corners of opaque objects.

• Single Slit Experiment: By holding two razor blades close together and viewing a straight bulb filament, you can easily observe diffraction. The lens of the eye focuses the pattern on the retina.
• Condition for Minima: $\theta = n\lambda/a$ ($n = \pm1, \pm2, \dots$).
• Condition for Secondary Maxima: $\theta = (n + 1/2)\lambda/a$.
• Central Maximum: The angular width is $2\theta = \frac{2\lambda}{a}$. The linear width on a screen at distance $D$ is $\frac{2\lambda D}{a}$.
• → [JEE TIP] Interference vs Diffraction: In YDSE, the pattern is technically a superposition of single-slit diffraction (the envelope) and double-slit interference (the fine fringes inside). All interference fringes are of equal intensity, but diffraction fringes rapidly decrease in intensity.
• → [JEE TIP] Feynman's Distinction: Richard Feynman noted there is no profound physical difference between interference and diffraction. It is a matter of usage: two sources is called "interference", while a large/infinite number of sources (like an entire slit) is called "diffraction".

Polarisation of Light

Light is a transverse electromagnetic wave; the electric field oscillates right angles to the direction of propagation. An unpolarised wave has its electric vector rapidly and randomly changing directions in the transverse plane.

• Polaroids: Sheets consisting of long-chain molecules. They transmit only the electric field component perpendicular to the molecules (the pass-axis). Unpolarised light passing through a polaroid halves in intensity ($I_0 = I_{in}/2$).
• Malus's Law: Transmitted intensity between two polaroids at angle $\theta$ is $I = I_0 \cos^2\theta$.
• → [JEE TIP] Brewster's Law: When unpolarised light strikes a boundary between two transparent media, the reflected light is completely plane-polarised if the angle of incidence is the Brewster angle $i_p$. This occurs when the reflected and refracted rays are strictly orthogonal ($90^\circ$ apart). The law is $\tan i_p = \mu_{21}$.
• → [JEE TIP] Trap 6 - Three Polaroids: If three polaroids $P_1, P_2, P_3$ are placed, and $P_1$ & $P_3$ are crossed ($90^\circ$), no light emerges. But if $P_2$ is placed between them at angle $\theta$, the final intensity is $I = I_0 \cos^2\theta \sin^2\theta = (I_0/4)\sin^2(2\theta)$. Maximum intensity $I_0/4$ is achieved at $\theta = 45^\circ$.

Conditions & Limitations

1. Paraxial Approximation: All mirror and lens derivations ($\frac{1}{v}\pm\frac{1}{u}=\frac{1}{f}$) strictly assume paraxial rays (rays making very small angles with the principal axis). Non-paraxial rays suffer from spherical aberration.
2. Thin Lens Approximation: The lens formula and combinations assume the thickness of the lens is negligible, treating optical centers as coincident.
3. YDSE Linear Fringes: The uniform fringe spacing formula $\Delta x = y d/D$ heavily relies on the assumption that $D \gg d$, and the angular spread is extremely small ($\sin\theta \approx \tan\theta \approx \theta$).
4. Independence of Coherent Sources: Two independent light sources (like two different bulbs) can never produce a steady interference pattern because their relative phase $\phi$ changes randomly on the order of $10^{-10}$ seconds.
5. Telescope Limitation: Real telescopes suffer from diffraction limits. No matter how perfect the lens, diffraction of the finite aperture limits the resolution.

Standard Derivations & Step-by-Step Problem Solving

• Mirror Equation Derivation: Draw a ray parallel to the axis passing through $F$, and a ray striking the pole $P$ at angle $i=r$. Using similar right triangles $A'B'F \sim MPF$ and $A'B'P \sim ABP$, we equate ratios. $\frac{B'A'}{BA} = \frac{B'F}{FP}$ and $\frac{B'A'}{BA} = \frac{B'P}{BP}$. Equating gives $\frac{B'F}{FP} = \frac{B'P}{BP}$. Applying sign conventions yields $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$.
• Refraction at Spherical Surface Derivation: Draw normal through $C$. For a ray hitting at $N$, the exterior angle of the triangle is $i = \angle NOM + \angle NCM = \frac{MN}{OM} + \frac{MN}{MC}$. Similarly, $r = \angle NCM - \angle NIM$. Applying Snell's Law for small angles ($n_1 i = n_2 r$) and substituting the angles gives $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2-n_1}{R}$.
• Lens Maker's Formula Strategy: Treat refraction at the two surfaces as a two-step process.
  1. First surface forms an image $I_1$: $\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2-n_1}{R_1}$.
  2. This image $I_1$ acts as a virtual object for the second surface: $\frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1-n_2}{R_2}$. Adding the equations cancels the intermediate virtual object distance $v_1$.

Important Graphs & Diagrams

• Angle of Deviation ($\delta$) vs Angle of Incidence ($i$): A parabolic-like asymmetric curve for a prism. For any given value of $\delta$ (except $D_m$), there are two corresponding angles of incidence ($i$ and $e$). At the absolute minimum $D_m$, $i = e$.
• Intensity Distribution in YDSE vs Single Slit:
  • YDSE: A series of identical sinusoidal peaks of equal maximum intensity ($4I_0$) evenly spaced across the screen.
  • Single Slit Diffraction: A massive central peak containing most of the energy, surrounded by alternating secondary maxima whose intensities rapidly diminish towards zero ($I_{central} \gg I_{1st} \gg I_{2nd}$).
• Huygens' Construction Diagram: When a plane wave strikes a denser medium at angle $i$, we construct spheres of radius $v_2 \tau$ from the interface. The forward envelope of these spheres gives the new refracted plane wave, physically demonstrating why the wave bends toward the normal and why the wavelength shrinks.

Previous Year JEE Topics

• Shift of Fringes in YDSE: Insertion of glass/mica sheets and identifying the central maximum's new position.
• Total Internal Reflection: Complex geometry problems involving prisms and TIR at multiple faces to trace the emergent ray.
• Lens Maker's Formula: Calculating the focal length when a lens is immersed in a liquid (often where $n_{liquid} > n_{lens}$, turning a convex lens diverging).
• Silvered Lenses: Finding the effective focal length and image position when one face of a plano-convex lens is silvered.
• Displacement Method: Calculating focal length or object size $O = \sqrt{I_1 I_2}$ from the two image sizes.
• Intensity Superposition: Calculating resultant intensity $I_R = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$ when partially coherent waves or unpolarised lights are mixed via polaroids.
• Three Polaroids Setup: Calculating resultant intensity $I = (I_0/4)\sin^2(2\theta)$ when unpolarised light passes through 3 polaroids with the middle one rotated.
• Combinations of Lenses/Mirrors: Finding the effective focal length when a lens is immersed in a liquid or silvered, and treating intermediate real images as virtual objects for subsequent lenses.
• Wavefront Geometry: Identifying the shape of wavefronts (e.g., light emerging from a convex lens with a source at focus becomes a plane wavefront).

JEE Traps

1. Trap: Image Velocity in Spherical Mirrors

• Misconception $\rightarrow$ An object moving towards a spherical mirror at a constant speed produces an image that also moves at a constant speed.
• Correct Understanding $\rightarrow$ The image's velocity is non-linear. As the object approaches the mirror, the average speed of the image increases substantially. The longitudinal velocity formula is $v_{image} = -m^2 v_{object}$, meaning image speed depends heavily on instantaneous magnification.

2. Trap: Optical vs. Mass Density

• Misconception $\rightarrow$ A medium that is optically denser (higher refractive index) must inherently possess a higher mass density (mass per unit volume).
• Correct Understanding $\rightarrow$ Optical density strictly refers to the ratio of the speed of light in two media and has no direct correlation with mass density. For example, turpentine has a lower mass density than water but is optically denser (it slows light down more).

3. Trap: Wave Energy During Refraction

• Misconception $\rightarrow$ When light enters a denser medium and its propagation speed decreases, the energy carried by the light wave also decreases.
• Correct Understanding $\rightarrow$ The energy carried by a wave depends on the amplitude and frequency of the wave, not on its speed of propagation. The energy is conserved even though the wave slows down.

4. Trap: Frequency Change Upon Refraction

• Misconception $\rightarrow$ The frequency of light changes when it refracts into a new medium because its speed and wavelength change.
• Correct Understanding $\rightarrow$ Frequency strictly remains constant. Atoms in the medium act as forced oscillators that take up the frequency of the external incident light. Only the speed and wavelength change ($\lambda_2 = \lambda_1 \frac{v_2}{v_1}$).

5. Trap: The "Half-Covered" Lens or Mirror

• Misconception $\rightarrow$ Covering half of a spherical mirror or lens with opaque black paper will result in only half of the image being formed.
• Correct Understanding $\rightarrow$ The entire, complete image is still formed by the rays hitting the remaining exposed optical surface. However, because the total area of the reflecting/refracting surface is reduced, the intensity (brightness) of the image is proportionally reduced.

6. Trap: The "Floating" Real Image

• Misconception $\rightarrow$ A real image does not actually exist unless there is a physical screen placed at the image distance to catch it.
• Correct Understanding $\rightarrow$ The real image genuinely exists suspended in space exactly where the refracted/reflected rays converge. A screen is merely used to diffuse those converging rays so that they can reach our eyes from various viewing angles.

7. Trap: Power Units in Lens Combinations

• Misconception $\rightarrow$ To find the equivalent power $P = P_1 + P_2$, you can simply calculate $1/f_1 + 1/f_2$ using the focal lengths exactly as given, even if they are in centimeters.
• Correct Understanding $\rightarrow$ Power is expressed in Dioptres (D) only if the focal length is specifically converted to meters. You must use $P = 1/f (\text{in meters})$ or strictly use $P = 100/f (\text{in cm})$ before adding powers together.

8. Trap: Incoherent Sources in YDSE

• Misconception $\rightarrow$ Two identical, independent light sources (like two distinct sodium lamps) placed side-by-side will produce a steady interference pattern.
• Correct Understanding $\rightarrow$ Independent sources are completely incoherent because they undergo abrupt, random phase changes every $\approx 10^{-10}$ seconds. This rapid phase shifting prevents fixed fringes from forming, resulting instead in a time-averaged uniform intensity ($I = 2I_0$) everywhere on the screen.

9. Trap: The Limit of Refraction (Total Internal Reflection)

• Misconception $\rightarrow$ Refraction always occurs to some degree when light travels from a denser medium to a rarer medium, regardless of the angle.
• Correct Understanding $\rightarrow$ Refraction into the rarer medium is only possible if the angle of incidence is strictly less than the critical angle ($i < i_c$). For any angle $i > i_c$, Snell's law cannot be satisfied, no refracted wave exists, and the light is totally internally reflected.

10. Trap: The Three Polaroids Setup

• Misconception $\rightarrow$ If two polaroids are crossed ($90^\circ$ apart) so no light transmits, placing a third polaroid between them will still result in zero transmitted light.
• Correct Understanding $\rightarrow$ The inserted middle polaroid actually rotates the plane of polarisation of the light. Following Malus's Law twice, the final transmitted intensity becomes $I = (I_0/4)\sin^2(2\theta)$. This allows light to pass through, reaching a maximum intensity of $I_0/4$ when the middle polaroid is at exactly $\theta = 45^\circ$.