Part IV, Chapter 4

Fiber Optics

Optical fibers guide light through total internal reflection, forming the backbone of modern telecommunications. Understanding their modal structure, dispersion, and amplification is essential for photonics engineering.

4.1 Total Internal Reflection and Waveguiding

An optical fiber consists of a core (refractive index n₁) surrounded by a cladding (n₂ < n₁). Light entering within the acceptance cone undergoes total internal reflection at the core-cladding interface and is guided along the fiber.

4.1.1 Numerical Aperture

The maximum acceptance half-angle θ_a defines the numerical aperture:

$$\text{NA} = \sin\theta_a = \sqrt{n_1^2 - n_2^2} \approx n_1\sqrt{2\Delta}$$

where Δ = (n₁ - n₂)/n₁ is the relative index difference. Typical telecom fibers have NA ≈ 0.12 and Δ ≈ 0.3%.

4.1.2 Step-Index vs. Graded-Index Fibers

Step-index fiber:

Uniform core index with an abrupt change at the cladding. Simple construction but suffers from intermodal dispersion in multimode fibers, as higher-order modes travel longer paths.

Graded-index fiber:

Core index decreases parabolically from center to edge: n(r) = n₁(1 - Δ(r/a)²). Higher-order modes travel longer paths but in lower-index regions (faster), nearly equalizing transit times. Reduces intermodal dispersion by orders of magnitude.

4.2 Guided Modes and the V-Number

The number of guided modes in a step-index fiber is determined by the normalized frequency parameter (V-number):

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

Derivation: V-Number from the Eigenvalue Equation

Step 1. The wave equation in cylindrical coordinates for a step-index fiber separates into Bessel functions in the core and modified Bessel functions in the cladding:

$$\text{Core: } J_l(\kappa r), \quad \text{Cladding: } K_l(\gamma r)$$

where κ² = n₁²k² - β² and γ² = β² - n₂²k².

Step 2. Matching boundary conditions at r = a gives the characteristic equation. The normalized parameters are:

$$U = \kappa a, \quad W = \gamma a, \quad V^2 = U^2 + W^2$$

Step 3. The fiber is single-mode when V < 2.405 (the first zero of J₀). For V >> 1, the number of guided modes is approximately:

Result:

$$M \approx \frac{V^2}{2} \quad \text{(step-index)}, \qquad M \approx \frac{V^2}{4} \quad \text{(graded-index)}$$

4.2.1 Single-Mode Fiber

Standard single-mode fiber (SMF-28) has a = 4.1 μm, Δ = 0.36%, giving V = 2.17 at λ = 1550 nm. Only the HE₁₁ (LP₀₁) mode propagates. The mode field diameter (MFD) is approximately:

$$\frac{w}{a} \approx 0.65 + \frac{1.619}{V^{3/2}} + \frac{2.879}{V^6}$$

(Marcuse approximation). For SMF-28, MFD ≈ 10.4 μm at 1550 nm.

4.3 Chromatic Dispersion

Chromatic dispersion causes different wavelength components of a pulse to travel at different speeds, broadening the pulse. It has two contributions:

4.3.1 Material Dispersion

The refractive index of silica depends on wavelength. The group velocity dispersion (GVD) parameter is:

$$\beta_2 = \frac{d^2\beta}{d\omega^2} = -\frac{\lambda^2}{2\pi c} D$$

where D is the dispersion parameter in ps/(nm·km). For standard silica fiber, D = 0 at λ ≈ 1310 nm (zero-dispersion wavelength). At 1550 nm, D ≈ 17 ps/(nm·km).

4.3.2 Waveguide Dispersion

The effective index of the guided mode depends on wavelength through the mode confinement. Waveguide dispersion is negative at telecom wavelengths and can be engineered (dispersion-shifted fibers) to move the zero-dispersion wavelength to 1550 nm. The dispersion length is:

$$L_D = \frac{T_0^2}{|\beta_2|}$$

where T₀ is the initial pulse width. For T₀ = 10 ps at 1550 nm, L_D ≈ 5 km.

4.4 Loss Mechanisms and Fiber Amplifiers

4.4.1 Fiber Loss

The attenuation of light in a fiber follows Beer's law: P(z) = P(0) exp(-αz), where α is the loss coefficient. In dB/km:

$$\alpha_{\text{dB}} = -\frac{10}{L} \log_{10}\!\left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) = 4.343\,\alpha$$

The main loss mechanisms are Rayleigh scattering (∝ λ^-4) at short wavelengths and infrared absorption at long wavelengths. The minimum loss of 0.2 dB/km occurs at λ ≈ 1550 nm, setting the optimal telecom window.

4.4.2 Erbium-Doped Fiber Amplifier (EDFA)

The EDFA revolutionized fiber telecommunications by providing optical amplification directly in the fiber, eliminating the need for electronic regeneration. Erbium ions (Er³+) doped into silica fiber provide gain in the 1530-1565 nm band (C-band) when pumped at 980 nm or 1480 nm.

$$G_{\text{dB}} = 10\log_{10}\!\left(\frac{P_{\text{out}}}{P_{\text{in}}}\right) = (g - \alpha) L \cdot 4.343$$

Typical EDFA gains are 20-40 dB with noise figures of 4-6 dB. The noise figure is limited by amplified spontaneous emission (ASE) with a quantum limit of 3 dB.

4.4.3 Wavelength Division Multiplexing (WDM)

WDM transmits multiple wavelength channels simultaneously through a single fiber. Dense WDM (DWDM) systems use 40-80+ channels spaced by 100 GHz (0.8 nm) or 50 GHz across the C-band. Combined with EDFAs, this enables aggregate data rates exceeding 10 Tbps per fiber.

4.5 Python Simulation: Fiber Modes, Dispersion & Loss

This simulation computes the V-number and number of modes, plots the fiber loss spectrum, shows pulse broadening due to dispersion, and displays the refractive index profiles.

Fiber Optics: Modes, Dispersion & Loss

Python

script.py124 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import jv as Jv  # Bessel function

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# --- Plot 1: Number of modes vs V-number ---
ax1 = axes[0, 0]
V_vals = np.linspace(0.5, 20, 200)
M_step = V_vals**2 / 2
M_graded = V_vals**2 / 4

ax1.plot(V_vals, M_step, color='#f59e0b', linewidth=2, label='Step-index (V^2/2)')
ax1.plot(V_vals, M_graded, color='#fb923c', linewidth=2, linestyle='--', label='Graded-index (V^2/4)')
ax1.axvline(2.405, color='#fbbf24', linestyle=':', alpha=0.7, label='V=2.405 (single-mode cutoff)')
ax1.set_xlabel('V-number', color='white')
ax1.set_ylabel('Number of modes', color='white')
ax1.set_title('Guided Modes vs V-Number', color='#fbbf24', fontsize=13)
ax1.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax1.set_ylim(0, 200)
ax1.tick_params(colors='white')
for spine in ax1.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 2: Fiber loss spectrum ---
ax2 = axes[0, 1]
lam = np.linspace(800, 1700, 500)  # nm

# Rayleigh scattering (~ lambda^-4)
alpha_rayleigh = 0.7 * (1000/lam)**4  # dB/km, normalized

# IR absorption (exponential tail)
alpha_ir = 0.01 * np.exp((lam - 1000) / 200)

# OH absorption peak at 1385 nm
alpha_oh = 0.5 * np.exp(-((lam - 1385)/15)**2)

alpha_total = alpha_rayleigh + alpha_ir + alpha_oh

ax2.semilogy(lam, alpha_rayleigh, '--', color='#60a5fa', linewidth=1, alpha=0.7, label='Rayleigh')
ax2.semilogy(lam, alpha_ir, '--', color='#f87171', linewidth=1, alpha=0.7, label='IR absorption')
ax2.semilogy(lam, alpha_oh, '--', color='#a78bfa', linewidth=1, alpha=0.7, label='OH peak')
ax2.semilogy(lam, alpha_total, '-', color='#f59e0b', linewidth=2.5, label='Total')
ax2.axvspan(1530, 1565, alpha=0.1, color='#f59e0b', label='C-band')
ax2.set_xlabel('Wavelength (nm)', color='white')
ax2.set_ylabel('Loss (dB/km)', color='white')
ax2.set_title('Optical Fiber Loss Spectrum', color='#fbbf24', fontsize=13)
ax2.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=8)
ax2.set_ylim(0.1, 10)
ax2.tick_params(colors='white')
for spine in ax2.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 3: Pulse broadening due to dispersion ---
ax3 = axes[1, 0]
z_vals = np.array([0, 5, 20, 50])  # km
T0 = 10e-12  # 10 ps initial pulse width
beta2 = -21.7e-27  # s^2/m for standard fiber at 1550 nm (D=17 ps/nm/km)

t = np.linspace(-100e-12, 100e-12, 1000)
for z_km in z_vals:
    z = z_km * 1e3  # convert to meters
    T_z = T0 * np.sqrt(1 + (z * beta2 / T0**2)**2)
    pulse = np.exp(-t**2 / (2 * T_z**2))
    pulse /= np.max(pulse)
    ax3.plot(t*1e12, pulse, linewidth=2, label=f'z = {z_km} km (T={T_z*1e12:.1f} ps)')

ax3.set_xlabel('Time (ps)', color='white')
ax3.set_ylabel('Normalized amplitude', color='white')
ax3.set_title('Pulse Broadening: GVD', color='#fbbf24', fontsize=13)
ax3.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=8)
ax3.tick_params(colors='white')
for spine in ax3.spines.values():
    spine.set_color('#f59e0b')

# --- Plot 4: Refractive index profiles ---
ax4 = axes[1, 1]
r = np.linspace(-1.5, 1.5, 500)  # normalized to core radius a
n1 = 1.4620
n2 = 1.4570

# Step-index
n_step = np.where(np.abs(r) <= 1, n1, n2)

# Graded-index (parabolic)
Delta = (n1 - n2) / n1
n_graded = np.where(np.abs(r) <= 1,
                     n1 * np.sqrt(1 - 2*Delta*(r)**2),
                     n2)

ax4.plot(r, n_step, color='#f59e0b', linewidth=2, label='Step-index')
ax4.plot(r, n_graded, color='#fb923c', linewidth=2, linestyle='--', label='Graded-index')
ax4.axvspan(-1, 1, alpha=0.05, color='#f59e0b')
ax4.annotate('Core', (0, n1+0.0003), ha='center', color='#fbbf24', fontsize=10)
ax4.annotate('Clad', (-1.3, n2+0.0003), ha='center', color='gray', fontsize=9)
ax4.annotate('Clad', (1.3, n2+0.0003), ha='center', color='gray', fontsize=9)
ax4.set_xlabel('r / a', color='white')
ax4.set_ylabel('Refractive index n(r)', color='white')
ax4.set_title('Fiber Index Profiles', color='#fbbf24', fontsize=13)
ax4.legend(facecolor='#1a1a2e', edgecolor='#f59e0b', labelcolor='white', fontsize=9)
ax4.tick_params(colors='white')
for spine in ax4.spines.values():
    spine.set_color('#f59e0b')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.show()

# Numerical examples
a = 4.1e-6    # core radius (m)
lam_op = 1550e-9
NA = np.sqrt(n1**2 - n2**2)
V = 2*np.pi*a/lam_op * NA
print(f"SMF-28 at 1550 nm:")
print(f"  n1 = {n1}, n2 = {n2}, Delta = {Delta*100:.3f}%")
print(f"  NA = {NA:.4f}")
print(f"  V = {V:.3f} ({'single-mode' if V < 2.405 else 'multimode'})")
print(f"  MFD ~ {2*a*1e6*(0.65 + 1.619/V**1.5 + 2.879/V**6):.1f} um")
print(f"\nDispersion length for T0=10 ps:")
print(f"  L_D = {T0**2/abs(beta2)/1e3:.1f} km")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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