(a) In a step-index optical fiber system, explain the terms pulse dispersion and material dispersion. An optical fiber having…

(a) Pulse dispersion and material dispersion in a step-index fibre

Pulse dispersion is the broadening in time of an optical pulse as it travels along the fibre. In a step-index multimode fibre, rays at different angles to the axis travel different path lengths, so they arrive at different times (intermodal dispersion). Material dispersion arises because the core's refractive index depends on wavelength, so the spectral components of a source of width Δλ travel at different group speeds.

Pulse (intermodal) dispersion for L = 1 km: the axial ray takes t₁ = n₁L/c. The steepest guided ray (cos θ = n₂/n₁) has path length n₁L/n₂ and takes t₂ = n₁²L/(c n₂). So

Δτ = t₂ − t₁ = n₁L(n₁ − n₂)/(c n₂) = 1.463 × 1000 × 0.019/(3×10⁸ × 1.444) = 6.42×10⁻⁸ s ≈ 64 ns.

(In the weak-guidance approximation Δτ ≈ L(n₁ − n₂)/c ≈ 63 ns.)

Material dispersion: Δτ_m = D_m·L·Δλ = 18.23 ps/(km·nm) × 1 km × 2 nm = 36.5 ps.

Modal dispersion (about 64 ns) is therefore about 1700 times larger than material dispersion (36.5 ps), which is negligible here, and the total spread is effectively 64 ns.

(b) Chromatic aberration and achromatism of two thin lenses in contact

Chromatic aberration is the defect in which a lens focuses different colours at different points, because its refractive index and so its focal length depend on wavelength; for a single converging lens, blue is focused nearer than red, so images show coloured fringes.

For two thin lenses in contact the power is P = 1/f₁ + 1/f₂. For a lens of dispersive power ω = (n_F − n_C)/(n_D − 1), the change in power between F and C light is δP = ωP. The combination is achromatic when δP₁ + δP₂ = 0:

ω₁/f₁ + ω₂/f₂ = 0.

So the two lenses must have focal lengths of opposite sign (one convex, one concave) with f₁/f₂ = −ω₁/ω₂.

If both are of the same material, ω₁ = ω₂ and the condition gives 1/f₁ + 1/f₂ = 0, i.e. net power zero and f = ∞. The system is then afocal and cannot act as a useful achromatic doublet. Two lenses of the same material can be achromatic only when separated, by d = (f₁ + f₂)/2, as in an eyepiece, and not when in contact.

(c) Relativistic mass, momentum and collision

(i) γ = 1/√(1 − 0.64) = 5/3. Mass m = γm₀ = (5/3)(1.67×10⁻²⁷) = 2.78×10⁻²⁷ kg. Momentum p = mv = 2.78×10⁻²⁷ × 0.8 × 3×10⁸ = 6.68×10⁻¹⁹ kg m/s.

When the proton sticks to the stationary nucleus of mass M = 5.0×10⁻²⁶ kg, momentum and total energy are conserved, so the resultant has relativistic mass γm₀ + M and

V = p/(γm₀ + M) = 6.68×10⁻¹⁹/(5.278×10⁻²⁶) = 1.27×10⁷ m/s ≈ 0.042c, in the original direction.

(ii) K = E/2 means (γ − 1)m₀c² = γm₀c²/2, so γ = 2. The (relativistic) mass is m = γm₀ = 2m₀, twice the rest mass; a numerical value needs m₀, which is not given.

From 1/√(1 − v²/c²) = 2: v²/c² = 3/4, so v = (√3/2)c = 0.866c = 2.6×10⁸ m/s.