(a) Write conditions for working of a step-index optical fiber. In a step-index fiber, the core and cladding materials have…

(a) Step-index fibre: working conditions and fibre parameters

Conditions: the core index must exceed the cladding index (n₁ > n₂); light must enter the end face within the acceptance cone, so that at the core-cladding interface it is incident at more than the critical angle and is totally internally reflected; the interface must be smooth and absorption and scattering small. Given n₁ = 1.50, n₂ = 1.43. (i) Critical angle at the interface: sin θc = n₂/n₁ = 0.9533, so θc = 72.4°. The critical propagation angle (the largest angle a guided ray can make with the axis) is θp = 90° − θc = 17.6°; equivalently cos θp = n₂/n₁. (ii) Acceptance angle: n₀ sin θa = √(n₁² − n₂²), so NA = √(2.25 − 2.0449) = √0.2051 = 0.453 and θa = sin⁻¹(0.453) = 26.9° (air, n₀ = 1). (iii) The axial ray takes t_min = Ln₁/c. The extreme ray, at θp to the axis, travels L/cos θp = Ln₁/n₂, so t_max = Ln₁²/(cn₂). Δt = t_max − t_min = (Ln₁/c)(n₁/n₂ − 1). For L = 1 km: Ln₁/c = (10³ × 1.50)/(3×10⁸) = 5.0 μs, and n₁/n₂ − 1 = 0.0490, so Δt = 0.245 μs = 245 ns. (iv) The delay is proportional to length, so for 50 km: Δt = 50 × 0.245 μs = 12.2 μs.

(b) Streamline flow and the energy components per unit volume

Streamline flow: a steady flow in which every particle passing a given point follows the same path with the same velocity, so that streamlines, whose tangents give the velocity direction, do not cross and remain fixed in time. Continuity: ∂ρ/∂t + ∇·(ρv) = 0; for steady flow of an isotropic incompressible fluid ρ is constant, so ∇·v = 0 and A₁v₁ = A₂v₂ along a tube of flow. Take a streamline tube with the fluid non-viscous. In time dt, a volume dV = A₁v₁dt = A₂v₂dt enters at section 1 (pressure P₁, height h₁, speed v₁) and the same volume leaves at section 2. Work done by the pressure forces: (P₁ − P₂)dV. Change in kinetic energy: ½ρdV(v₂² − v₁²). Change in potential energy: ρgdV(h₂ − h₁). Work-energy: (P₁ − P₂) = ½ρ(v₂² − v₁²) + ρg(h₂ − h₁), so P + ½ρv² + ρgh = constant along the streamline. The total energy per unit volume therefore has three parts: pressure (flow) energy P, kinetic energy ½ρv², and potential energy ρgh, each with the dimensions of pressure (J m⁻³). This is Bernoulli's theorem.

(c)(i) Fresnel and Fraunhofer diffraction

Fresnel: the source or screen, or both, are at a finite distance from the obstacle; the wavefronts are spherical or cylindrical; no lens is needed; the analysis uses Fresnel half-period zones; the pattern is complicated and the central region may be bright or dark. Fraunhofer: both source and screen are effectively at infinity, usually achieved with two lenses; the wavefronts are plane; the pattern is described by the Fourier transform of the aperture; simpler maxima and minima, as in single slit and grating.

(c)(ii) Resolving power of a telescope and the microscope with UV light

The resolving power of a telescope is its ability to form separate images of two close point objects, measured as the reciprocal of the smallest resolvable angle. By Rayleigh's criterion, dθ = 1.22λ/D for an aperture of diameter D, so the resolving power is D/(1.22λ), increased by a larger objective and a shorter wavelength. For a microscope, the smallest resolvable separation of two points is d = 0.61λ/(n sin θ) = 0.61λ/NA, so the resolving power is 1/d = NA/(0.61λ), inversely proportional to the wavelength. Ultraviolet light has a smaller wavelength than visible light (about 250 nm against 550 nm), so d is smaller and the resolving power more than twice as great. The images are recorded on a photographic plate or detector, since the eye does not see UV, and the lenses must be of quartz.