Optical Fibre Communication (EC-801) - Unit 5 Short Notes

UNIT 5: OPTICAL FIBRE COMMUNICATION – SHORT NOTES

1. FIBER OPTICS FUNDAMENTALS

Numerical Aperture (NA)

• Definition: A measure of the light-gathering ability of an optical fiber. It defines the maximum angle of incidence at the fiber input face for which light will be guided.

• Derivation (Step-Index Fiber):

  For light entering from air (n₀=1) into a fiber core (n₁) surrounded by cladding (n₂), the condition for Total Internal Reflection (TIR) at the core-cladding interface is:

$$n_1 \sin \theta_1 \geq n_2$$

Using Snell's law at the air-core interface: $$\displaystyle \sin \theta_0 = n_1 \sin \theta_1 $$.

Combining and solving for the maximum acceptance angle $$\displaystyle \theta_{0(max)} $$:

$$n_0 \sin \theta_{0(max)} = \sqrt{n_1^2 - n_2^2}$$

Therefore, **Numerical Aperture (NA) is:**

$$\boxed{NA = \sqrt{n_1^2 - n_2^2}}$$

For a medium other than air (refractive index n₀), the maximum acceptance angle is given by:

$$\sin \theta_{0(max)} = \frac{\sqrt{n_1^2 - n_2^2}}{n_0}$$

• Critical Angle ($$\displaystyle \theta_c $$): The angle of incidence at the core-cladding interface beyond which TIR occurs.

$$\boxed{\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)}$$

• Relation: $$\displaystyle NA = n_1 \sqrt{2\Delta} $$ (for small $\Delta$, where $$\displaystyle \Delta = \frac{n_1 - n_2}{n_1} $$).

[!TIP] Exam Focus: NA derivation is very frequent. Remember NA depends only on n₁ and n₂. For a fiber in a medium other than air, NA is defined as $$\displaystyle n_0 \sin \theta_{0(max)} $$.

Modes of Propagation

• Mode: A specific solution to Maxwell's equations satisfying boundary conditions, representing a distinct path of light propagation.

• Planar Dielectric Waveguide: Supports Transverse Electric (TE) and Transverse Magnetic (TM) modes. Modes are designated by the number of field zeros (m for TE/TM).

• Step-Index Fiber (Cylindrical): Modes are hybrid (HE/EH). The exact solution involves Bessel functions. The number of guided modes (M) is approximately:

$$\boxed{M \approx \frac{V^2}{2}} \quad \text{(for large V, step-index)}$$

• Normalized Frequency (V-number): A dimensionless parameter determining the number of modes.

$$\boxed{V = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2} = \frac{2\pi a}{\lambda} \cdot NA}$$

where `a` = core radius, `λ` = operating wavelength.

*   **Significance:**

    *   V < 2.405 → **Single-mode operation** (only fundamental mode HE₁₁ propagates).

    *   V >> 2.405 → **Multimode operation**.

• Graded-Index Fiber: Number of modes is also proportional to V² but with a different constant. For a parabolic profile, $$\displaystyle M \approx \frac{g}{g+2} \frac{V^2}{2} $$ (g = profile parameter).

[!TIP] Common Pitfall: The exact cutoff for single-mode is V=2.405. For V slightly above this, a second mode (LP₁₁) may leak (leaky mode). Use V < 2.405 for guaranteed single-mode.

Ray Theory of Light Transmission

• Step-Index Fiber:

  • Meridional Rays: Pass through the fiber axis. Simple analysis using TIR.

  • Skew Rays: Do not pass through the axis; follow a helical path. They have a longer path length than meridional rays with the same θ₀.

  • Axial Rays: θ₀ = 0, travel straight along the axis.

  • Acceptance Cone: Light entering within the cone of half-angle $$\displaystyle \theta_{0(max)} $$ is guided. Solid angle of acceptance = $$\displaystyle \pi (NA)^2 $$.

• Graded-Index Fiber: Ray path is continuously curved due to gradual index change. Rays follow sinusoidal paths. This reduces intermodal dispersion as higher-order rays travel faster in the lower-index outer core.

Dispersion

• Definition: Broadening of an optical pulse as it travels along the fiber, limiting the maximum data rate.

• Intermodal (Modal) Dispersion: Occurs in multimode fibers due to different group velocities of different modes.

  • Step-Index: Very high. Pulse broadening $$\displaystyle \Delta \tau_{inter} \approx \frac{L n_1 \Delta}{c} $$.

  • Graded-Index: Greatly reduced. For ideal parabolic profile, $$\displaystyle \Delta \tau_{inter} \approx 0 $$.

• Intramodal (Chromatic) Dispersion: Occurs in all fibers (single-mode and multimode) because different wavelengths travel at different speeds.

  • Material Dispersion: Due to wavelength dependence of refractive index (n). Zero-dispersion wavelength for silica ~1.27 µm.

  • Waveguide Dispersion: Due to wavelength dependence of the mode propagation constant. Arises from the waveguide structure itself. Can be engineered to shift zero-dispersion wavelength.

  • Total Chromatic Dispersion: $$\displaystyle D_{total} = D_{material} + D_{waveguide} $$ (units: ps/(nm·km)).

• Polarization Mode Dispersion (PMD): In single-mode fibers, due to birefringence (asymmetry/stress), the two orthogonal polarization states travel at slightly different speeds. A random, statistical effect. $$\displaystyle \Delta \tau_{PMD} = PMD_{coeff} \cdot \sqrt{L} $$.

• Pulse Broadening & Bandwidth: Total RMS pulse broadening: $$\displaystyle \Delta \tau_{total} = \sqrt{\Delta \tau_{inter}^2 + \Delta \tau_{chrom}^2 + \Delta \tau_{PMD}^2} $$.

  • Bandwidth-Distance Product (BL): $$\displaystyle B \cdot L \approx \frac{0.44}{\Delta \tau_{total}} $$ (for NRZ). A key system performance metric.

• Rise Time Budget: For a link with multiple sections (transmitter, fiber, receiver), total RMS rise time:

$$\boxed{\Delta t_{r(system)} = \sqrt{\Delta t_{r(tx)}^2 + \Delta t_{r(fiber)}^2 + \Delta t_{r(rx)}^2}}$$

where $$\displaystyle \Delta t_{r(fiber)} = \Delta \tau_{total} $$.

[!TIP] Key Formula: For step-index multimode fiber, intermodal dispersion limit: $$\displaystyle B \cdot L \leq \frac{0.44 c}{n_1 \Delta} $$. Chromatic dispersion limit: $$\displaystyle B \cdot L \leq \frac{0.44}{|D_{total}| \Delta \lambda} $$.

2. ATTENUATION AND LOSS MECHANISMS

Attenuation coefficient $\alpha$ (dB/km) = 10 log₁₀(P_in/P_out).

[!TIP] Critical Calculation: Critical Radius of Curvature ($$\displaystyle R_c $$) for large bending losses:

$$\boxed{R_c \approx \frac{3\lambda}{4\pi n_1 \sqrt{2\Delta}}}$$

Loss increases sharply when bend radius < $$\displaystyle R_c $$.

3. FIBER FABRICATION TECHNIQUES

[!TIP] Exam Comparison: MCVD is inside-out (deposits on tube inner wall),