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Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| Numerical Aperture (NA) | NA = sqrt(n1^2 - n2^2) = n1 x sqrt(2 x Delta) | n1 = core refractive index; n2 = cladding refractive index; Delta = (n1 - n2) / n1 = relative refractive index difference; NA determines acceptance cone half-angle: theta_a = arcsin(NA) |
| Attenuation in a Fiber Link | P_out = P_in x 10^(-alpha x L / 10) | P_out = output power; P_in = input power; alpha = attenuation coefficient (dB/km); L = fiber length (km); total loss = alpha x L in dB |
| Cutoff Wavelength (Single-Mode Condition) | lambda_c = (pi x d x NA) / 2.405 | d = core diameter (m); NA = numerical aperture; 2.405 = first zero of Bessel function J_0; single-mode operation requires lambda > lambda_c |
| V-Number (Normalized Frequency) | V = (pi x d x NA) / lambda | d = core diameter; lambda = operating wavelength; NA = numerical aperture; for V < 2.405: single-mode propagation; number of modes approx V^2 / 2 for step-index multimode fiber |
| Material Dispersion | sigma_mat = |D_mat| x L x delta_lambda | D_mat = material dispersion coefficient (ps/nm/km); L = fiber length (km); delta_lambda = source spectral width (nm); sigma_mat = pulse broadening in ps |
| Optical Link Power Budget | P_tx - P_rx = alpha x L + sum(connector_losses) + sum(splice_losses) + margin | P_tx = transmitter power (dBm); P_rx = receiver sensitivity (dBm); alpha = fiber attenuation (dB/km); L = link length; margin = system margin (typically 3 to 6 dB) |
Key concepts
Single-Mode vs Multimode Fiber
Single-mode fiber (SMF) has a small core (8 to 10 um) and V < 2.405, carrying only the HE11 mode. It has very low dispersion and is used for long-haul links (tens to thousands of km). Multimode fiber has a larger core (50 or 62.5 um) and supports many modes, causing modal dispersion that limits bandwidth-distance product. Graded-index multimode fiber reduces modal dispersion by using a parabolic refractive index profile.
Attenuation Mechanisms
Rayleigh scattering (dominant below 1500 nm) is caused by sub-wavelength density fluctuations in the glass; scales as lambda^(-4). Absorption: intrinsic (UV and IR absorption tails of silica) and extrinsic (OH ion absorption peak near 1380 nm and metallic impurities). Bending loss: macro-bending (bend radius below critical value) and micro-bending (lateral stress or manufacturing imperfections in the core-cladding interface).
Dispersion Types
Modal dispersion: different modes travel at different speeds; significant in step-index multimode fiber; eliminated in single-mode fiber. Chromatic dispersion: combines material dispersion (refractive index varies with wavelength) and waveguide dispersion (mode field changes with wavelength). Polarization Mode Dispersion (PMD): two polarization states of HE11 mode travel at slightly different speeds due to fiber birefringence; a concern at 10 Gbps and above.
Optical Amplifiers (EDFA)
Erbium-Doped Fiber Amplifier (EDFA) amplifies signals in the 1530 to 1565 nm C-band by stimulated emission from Er3+ ions pumped at 980 nm or 1480 nm. It provides gain of 20 to 40 dB and amplifies all WDM channels simultaneously. Limitations include amplified spontaneous emission (ASE) noise, non-flat gain spectrum, and lack of dispersion compensation.
WDM and Optical Windows
Wavelength Division Multiplexing (WDM) transmits multiple channels on separate wavelengths over one fiber. O-band: 1260 to 1360 nm (legacy multimode, some PON). E-band: 1360 to 1460 nm (high OH absorption, avoided). S-band: 1460 to 1530 nm. C-band: 1530 to 1565 nm (lowest attenuation ~0.2 dB/km, EDFA gain window). L-band: 1565 to 1625 nm. DWDM uses channel spacing of 100 GHz or 50 GHz in C-band.
Coherent vs Direct Detection
Direct detection (IM-DD): photodetector converts optical power to current; simple, low cost, used for short-reach links. Coherent detection: mixes received signal with local oscillator laser; detects amplitude and phase; enables higher spectral efficiency modulation (QPSK, 16-QAM); requires digital signal processing (DSP) for dispersion and PMD compensation; standard for 100 Gbps and beyond long-haul links.
Tables
Optical Fiber Transmission Windows
| Window | Wavelength (nm) | Attenuation (dB/km) |
|---|---|---|
| O-band | 1260 to 1360 | ~0.35 |
| E-band | 1360 to 1460 | ~2.0 (OH peak) |
| S-band | 1460 to 1530 | ~0.25 |
| C-band | 1530 to 1565 | ~0.20 (minimum) |
| L-band | 1565 to 1625 | ~0.22 |
Fiber Type Comparison
| Parameter | SMF (G.652) | MMF 50 um |
|---|---|---|
| Core diameter | 8 to 10 um | 50 um |
| NA | 0.12 to 0.14 | 0.20 to 0.24 |
| Bandwidth x Distance | > 100 THz km | 2 GHz km (OM3) |
| Typical use | Long-haul, metro | Data centre (< 1 km) |
Quick facts
- Minimum attenuation of silica single-mode fiber is approximately 0.18 to 0.20 dB/km at 1550 nm.
- Single-mode fiber requires V < 2.405; for a standard G.652 fiber (d = 9 um, NA = 0.13), lambda_c is around 1260 nm.
- Rayleigh scattering loss scales as lambda^(-4); doubling wavelength reduces scattering loss by a factor of 16.
- Standard EDFA pump wavelengths are 980 nm (lower noise) and 1480 nm (higher output power).
- DWDM C-band channel spacing: 100 GHz corresponds to approximately 0.8 nm at 1550 nm.
- Connector loss is typically 0.3 to 0.5 dB per connector pair; splice loss is 0.05 to 0.1 dB per fusion splice.
- Bandwidth-distance product for graded-index 50 um multimode fiber (OM4) is 4700 MHz x km at 850 nm.
- PMD coefficient for modern SMF is less than 0.1 ps/sqrt(km); for a 400 km link, PMD = 0.1 x sqrt(400) = 2 ps.
Exam shortcuts
- V-number calculation: V = (pi x d x NA) / lambda. Always convert d to the same units as lambda. For d = 50 um, NA = 0.22, lambda = 850 nm: V = (pi x 50e-6 x 0.22) / 850e-9 = 40.6. Number of modes = V^2 / 2 = 824.
- Link power budget in 3 steps: (1) available power margin = P_tx - P_rx (dBm). (2) total loss = alpha x L + connector losses + splice losses. (3) verify margin = available - total loss > minimum system margin (3 dB). If margin is negative, the link will not work.
- Attenuation loss dB = 10 log10(P_in / P_out). If input is 1 mW (0 dBm) and output is 1 uW (-30 dBm), loss = 30 dB. Link length = 30 dB / 0.3 dB/km = 100 km. The dB approach avoids handling tiny power ratios.
- For dispersion limit on bit rate: maximum bit rate B less than or equal to 1 / (4 sigma_total), where sigma_total is total pulse broadening. If dispersion gives sigma = 0.1 ns over the link, B less than or equal to 1 / 0.4 ns = 2.5 Gbps.
- Acceptance angle from NA: theta_a = arcsin(NA). For NA = 0.22: theta_a = arcsin(0.22) = 12.7 degrees. The full acceptance cone (both sides) is 2 x 12.7 = 25.4 degrees.