Optics

In geometrical optics, image formation by spherical mirrors obeys the New Cartesian Sign Convention: all distances are measured from the pole (P) along the principal axis; distances in the direction of incident light are positive, while those opposite are negative; heights above the principal axis are positive, and those below are negative.

For a spherical mirror of radius of curvature R, the focal length is f = R / 2 (concave mirror: f < 0; convex mirror: f > 0). The paraxial ray derivation yields the fundamental Mirror Formula:

where u is object distance, v is image distance, and lateral magnification is defined as:

When light crosses a boundary separating optical medium 1 (index n₁) and medium 2 (index n₂), it obeys Snell's Law:

For an object in an optically denser medium (n > 1) viewed from air, refraction bends rays away from the normal, causing an apparent upward shift:

Total Internal Reflection (TIR): When light travels from an optically denser medium toward an optically rarer medium (n₁ > n₂), the refracted ray bends away from the normal. At a specific angle of incidence called the Critical Angle (θ_c), the angle of refraction reaches r = 90°:

For any angle of incidence i > θ_c, 100% of the incident radiant energy is reflected back into the denser medium with zero transmission loss. Engineering applications include optical communication fibers (cylindrical glass core of high index n₁ surrounded by lower-index cladding n₂), prism periscopes, and diamond brilliance.

Consider a convex spherical refracting interface of radius of curvature R separating media of refractive indices n₁ and n₂. Applying Snell's law in the small-angle paraxial approximation (sin θ ≈ tan θ ≈ θ):

This master relation governs image formation across all curved refracting surfaces and serves as the stepping stone to thin lenses.

A thin lens consists of a transparent medium bounded by two spherical surfaces of radii R₁ and R₂. By applying the single-surface refraction equation sequentially to both surfaces and eliminating the virtual intermediate image distance:

where n = n_lens / n_medium is the relative refractive index of the lens material. Combining this with the object-image relation produces the standard Thin Lens Formula:

Power of a Lens (P): The measure of a lens's ability to converge or diverge light rays is defined as P = 1 / f (with f measured in meters). The SI unit of power is the Diopter (D) (1 D = 1 m⁻¹). For two coaxial thin lenses placed in intimate contact:

A triangular prism has two plane refracting faces inclined at an angle A (the refracting angle or apex angle). When a monochromatic ray enters face 1 at incidence angle i₁ and emerges from face 2 at angle i₂, the internal refraction angles satisfy:

The net angular deviation δ suffered by the ray is:

Condition of Minimum Deviation: As angle of incidence i increases, deviation δ initially decreases, reaches a unique minimum value δ_m, and then increases. By symmetry and the principle of reversibility of light, minimum deviation occurs when the ray passes symmetrically through the prism:

Applying Snell's law at the first refracting face yields the renowned Prism Refractive Index Formula:

For a thin prism with apex angle A < 10°, small-angle approximation yields: δ ≈ (n - 1) A.

According to Cauchy's empirical relation n(λ) = A + B/λ², the refractive index of glass decreases with increasing wavelength. Since λ_red > λ_violet, it follows that n_violet > n_red. Consequently, violet light deviates the most and red light deviates the least, dispersing composite white light into its spectral colors (VIBGYOR).

• Angular Dispersion (θ): The angular difference between emergent extreme rays:
  θ = δ_V - δ_R = (n_V - n_R) A
• Dispersive Power (ω): The ratio of angular dispersion to the mean deviation (represented by yellow sodium line δ_Y):
  ω = (δ_V - δ_R) / δ_Y = (n_V - n_R) / (n_Y - 1)
  Dispersive power depends solely on the optical material and is completely independent of the apex angle A.

A compound microscope provides high angular magnification of microscopic biological specimens using two converging lens systems mounted coaxially in a tube:

1. Objective Lens: Very short focal length f_o and small aperture. Forms a real, inverted, magnified intermediate image A'B' just inside the focal point of the eyepiece.
2. Eyepiece Lens: Moderate focal length f_e and larger aperture. Acts as a simple magnifier to form an enlarged, virtual final image A''B''.

The total angular magnifying power is m = m_o × m_e:

• Case 1: Final image at Least Distance of Distinct Vision (D = 25 cm):
  m = -(v_o / u_o) · (1 + D / f_e) ≈ -(L / f_o) · (1 + D / f_e)
  where L is the separation between the second focal point of the objective and the first focal point of the eyepiece (optical tube length).
• Case 2: Normal Adjustment (Final image at infinity for relaxed eye):
  m = -(v_o / u_o) · (D / f_e) ≈ -(L / f_o) · (D / f_e)

An astronomical refracting telescope produces angular magnification of distant celestial objects:

• Objective: Large focal length f_o and large aperture to maximize light gathering and resolving power.
• Eyepiece: Short focal length f_e and smaller aperture.
• Normal Adjustment (Image at Infinity): The intermediate real image lies at the common focal plane (F'_o = F_e). The tube length is L = f_o + f_e, and the magnifying power is:
  m = -f_o / f_e
• Image at D (25 cm): m = -(f_o / f_e) · (1 + f_e / D) with tube length L = f_o + u_e.

Reflecting Telescopes (Cassegrain & Newtonian): Use a large concave paraboloidal primary mirror instead of a heavy glass objective lens. Key advantages: (1) Completely free of chromatic aberration; (2) Spherical aberration minimized by parabolic curvature; (3) Structural mechanical support from the entire back face allows giant multi-meter mirrors.

In physical wave optics, a Wavefront is defined as the continuous locus of all oscillating points in an optical medium having the exact same phase of vibration. The direction of light propagation (a light ray) is always strictly perpendicular to the wavefront at every point.

Formulated by Christiaan Huygens in 1678, this principle provides a geometrical construction for determining the shape and position of a wavefront at any subsequent instant:

1. Every point on a primary wavefront acts as an independent point source of secondary spherical disturbances called secondary wavelets, which radiate outward into the forward medium at the characteristic wave speed v.
2. The forward envelope (tangential surface bounding all secondary wavelets) at any later time t + Δt defines the new position and shape of the primary wavefront.

Proof of Reflection: Consider a plane wavefront AB incident at angle i on a plane reflecting surface MN. While wave point B advances to strike the mirror at C over time τ = BC / v, secondary wavelets from point A expand into a hemisphere of radius AD = v τ = BC. Constructing tangent plane CD from C creates two right-angled triangles ΔABC and ΔCDA sharing common hypotenuse AC and equal sides BC = AD. By RHS congruency, ∠BAC = ∠DCA, which proves:

Proof of Snell's Law: Let a plane wavefront enter medium 2 having wave speed v₂ from medium 1 with speed v₁. In time τ, point B travels distance BC = v₁ τ in medium 1, while secondary wavelets from A advance distance AD = v₂ τ in medium 2. From geometry:

Dividing the two equations yields Snell's law:

Critical Historical Significance: Newton's corpuscular theory predicted that light travels faster in optically denser media (v_water > v_air). Huygens' wave theory predicted that light travels slower in denser media (v = c / n < c). Foucault's direct 1850 laboratory measurement proved v_water < v_air, definitively validating the wave model.

When two light waves of identical frequency meet at a spatial coordinate, the resultant electric field vector is the vector sum of individual fields: 𝐄 = 𝐄₁ + 𝐄₂. Interference is the redistribution of radiant energy in space resulting from the superposition of two or more coherent light waves.

In 1801, Thomas Young achieved coherent illumination by splitting a single wavefront through two closely spaced parallel slits S₁ and S₂ separated by distance d. A viewing screen is placed at distance D (where D ≫ d). For an observation point P at distance y from the central axis:

Phase difference: Δφ = (2π / λ) · Δx = (2π / λ) · (y d / D).

• Constructive Interference (Bright Fringes / Maxima):
  Δx = n λ ⟹ y_n = n (λ D / d) (n = 0, ±1, ±2, ...)
  n = 0 gives the central zero-order bright fringe at y = 0.
• Destructive Interference (Dark Fringes / Minima):
  Δx = (2n - 1) (λ / 2) ⟹ y_n' = (2n - 1) · (λ D / 2d) (n = 1, 2, ...)

The Fringe Width (β) is defined as the spatial separation between any two successive bright fringes or any two successive dark fringes:

Notice that fringe width β is completely uniform across the entire interference field.

Intensity Formula: If the individual slits produce field amplitudes a₁ and a₂ (intensities I₁ and I₂), the resultant intensity is:

For identical slits with equal intensity I₁ = I₂ = I₀:

The average intensity across the fringes is ⟨I⟩ = (I_max + I_min) / 2 = 2 I₀, which exactly equals the sum of unsuperposed intensities (I₁ + I₂), proving that interference does not create or destroy energy, but merely redistributes radiant energy in space in full accordance with the First Law of Thermodynamics.

When a thin transparent sheet of thickness t and refractive index n is placed across the path of one slit beam, light travels in glass with reduced speed v = c/n. The effective optical path length increases by (n - 1) t. Consequently, the entire fringe pattern shifts laterally without changing the fringe width β:

Diffraction is the bending of light waves around sharp edges, corners, or narrow apertures and their spreading into regions of geometrical shadow. Diffraction effects are noticeable only when the size of the obstacle or aperture a is comparable to the wavelength of the light wave (a ~ λ). Because visible light has microscopic wavelengths (400–700 nm), everyday optical diffraction requires narrow microscopic slits.

• Fresnel Diffraction: Source of light and observation screen are placed at finite distances from the diffracting aperture; wavefronts are spherical or cylindrical; no focusing lenses required.
• Fraunhofer Diffraction: Source and screen are effectively at infinite distances; incident wavefronts are plane; convex lenses are used to collimate and focus beams.

Consider a monochromatic plane wave of wavelength λ incident normally on a single slit of width a. By dividing the slit aperture into continuous Huygens wavelet emitters and integrating across the slit, we obtain the conditions for diffraction fringes at diffraction angle θ:

• Condition for Diffraction Minima (Dark Fringes):
  a sin θ_n = n λ (n = ±1, ±2, ±3, ...)
  Physical Proof: For the first minimum (n = 1), path difference across the slit edges is a sin θ = λ. Dividing the slit into two equal halves (width a/2 each), every wavelet in the top half has a counterpart in the bottom half with path difference (a/2) sin θ = λ/2, producing complete pairwise destructive interference.
• Condition for Secondary Maxima:
  a sin θ_n' = (2n + 1) · (λ / 2) (n = 1, 2, 3, ...)

The central diffraction maximum lies between the first minimum on either side (θ = ± λ/a):

• Angular Half-Width: θ₀ = λ / a (radians).
• Total Angular Width: 2 θ₀ = 2 λ / a.
• Linear Width on Screen (at distance D):
  β₀ = 2 y₁ = 2 (λ D / a)
  The central maximum is twice as wide as any secondary maximum (which has width λD/a)!

According to Lord Rayleigh's Criterion, two point sources are regarded as just resolved when the central diffraction maximum of one source coincides exactly with the first diffraction minimum of the second source.

• For a circular aperture of diameter D_aperture, the angular limit of resolution is:
  dθ = 1.22 λ / D_aperture
• Resolving Power of a Telescope: Reciprocal of the angular resolution:
  RP_telescope = 1 / dθ = D_aperture / (1.22 λ)
  Increasing objective diameter D_aperture sharpens celestial resolution.

Interference and diffraction establish that light is a wave phenomenon, but they do not distinguish between longitudinal waves (like sound) and transverse waves. Polarization provides conclusive, irrefutable proof that light waves are strictly transverse electromagnetic waves.

• Unpolarized Light: Ordinary light emitted by incandescent sources (sunlight, candle flame, LEDs) where the electric field vector 𝐄 vibrates symmetrically in all random spatial planes perpendicular to the propagation axis.
• Plane (Linearly) Polarized Light: Light in which the electric vector vibrations are confined strictly to a single spatial plane containing the direction of wave propagation (the plane of vibration). The plane perpendicular to the plane of vibration is called the plane of polarization.

Modern polarizers utilize synthetic sheets containing long-chain polyvinylene polymers aligned in parallel arrays. When unpolarized light enters, electric field components parallel to the molecular chains are absorbed by electron conduction, while orthogonal components pass unhindered (dichroism).

When unpolarized light of intensity I_in passes through an ideal polarizer, exactly half of the unpolarized intensity is transmitted: I₁ = 1/2 I_in.

When this plane-polarized light subsequently encounters an analyzer whose transmission axis is inclined at an angle θ to the polarizer axis, the transmitted amplitude is E = E₀ cos θ. The resulting intensity obeys Malus's Law (discovered by Étienne-Louis Malus in 1809):

• When θ = 0° (parallel axes): cos 0° = 1 ⟹ I = I₀ (maximum transmission).
• When θ = 90° (crossed polarizers): cos 90° = 0 ⟹ I = 0 (complete extinction of light).

In 1811, Sir David Brewster discovered that when unpolarized light is incident on a transparent dielectric boundary (such as air-glass or air-water), the reflected light becomes 100% plane-polarized at a specific incidence angle called the Polarizing Angle or Brewster Angle (θ_p or i_p).

Condition for Complete Polarization: Complete polarization occurs when the reflected ray and refracted ray are mutually perpendicular to each other:

Applying Snell's law at the interface (n₁ sin θ_p = n₂ sin r):

For standard crown glass (n = 1.5): θ_p = tan⁻¹(1.5) ≈ 56.3°; for water (n = 1.33): θ_p = tan⁻¹(1.33) ≈ 53.1°. The reflected ray has its electric vector vibrating purely perpendicular to the plane of incidence (dots), while the refracted ray remains partially polarized.

• Polaroid Sunglasses: Glare from horizontal surfaces like wet roads, snow, and lakes is horizontally polarized by reflection. Polaroid glasses with vertical transmission axes extinguish this reflected glare completely.
• Liquid Crystal Displays (LCDs): Nematic liquid crystals rotate the polarization vector of backlight between crossed polarizers, controlling pixel brightness via voltage pulses.
• Photoelastic Stress Analysis: Transparent plastic mechanical models subjected to mechanical stress exhibit optical birefringence, revealing interior stress concentrations as colored fringes under polarized light.