Determine the numerical aperture of a step-index optical fiber from core and cladding refractive indices. Calculates acceptance angle, V-number for single/multimode determination, and number of modes.
Optical fibers guide light by total internal reflection at the boundary between a higher-index core and a lower-index cladding. Two parameters describe how the fiber behaves: the numerical aperture (NA), which sets the maximum acceptance angle for light entering the fiber, and the V-number (normalized frequency), which determines how many propagating modes the fiber supports. Together they classify a fiber as single-mode (one fundamental mode only, V < 2.405) or multimode (many modes, used for high-bandwidth short-distance applications).
This calculator returns NA, acceptance angle, V-number, and the approximate number of modes from the core and cladding refractive indices, core diameter, and operating wavelength. Telecom standards: single-mode (SMF-28) has core ~8.2 µm, cladding 125 µm, NA ≈ 0.14, used at 1310 nm and 1550 nm. Multimode (OM4) has core 50 µm, cladding 125 µm, NA ≈ 0.20, used at 850 nm. The numbers and conventions are different from microscopy or photography NA, but the underlying principle (TIR confinement, light acceptance cone) is shared.
For fiber coupling work, NA is the dominant design parameter. A focused beam must match the fiber's NA to couple efficiently — too wide a beam wastes light (overfilled), too narrow leaves some modes unfilled (underfilled). For fiber laser delivery, choose a fiber with high enough NA to handle the input beam. For telecom, single-mode is the standard for distances over ~1 km because modal dispersion in multimode fiber limits bandwidth.
**Scenario:** Need a multimode fiber for 100 Gbps Ethernet over 100 m at 850 nm with VCSEL laser. OM4 spec: core 50 µm, NA 0.20. **Calculation:** V = (π × 50 / 0.85) × 0.20 = 36.9. Number of modes ≈ V²/2 = 680 modes. Modal dispersion: ~250 ps/km (typical OM4). At 100 m, dispersion = 25 ps. At 100 Gbps (10 ps bit period), this dispersion is significant — uses 4×25G WDM or single-mode for higher reach. **Result:** OM4 multimode supports 100 Gbps over 100 m via parallel multimode (PMM, 4 × 25G fibers) or 4 × 25G WDM. Beyond 100 m, modal dispersion dominates; single-mode is required. For very short data center links (< 30 m), OM4 100G via parallel fibers is cost-effective.
**Scenario:** Couple a 980 nm diode laser into single-mode fiber (NA 0.14). Laser beam after collimation: D = 2 mm. What focal length lens? **Calculation:** Fiber NA 0.14 → acceptance half-angle θ = arcsin(0.14) = 8.05°. To focus the 2 mm beam into this cone with a lens: tan(8.05°) = 1/(2×f#) → f# = 1/(2 × tan(8.05°)) = 3.54. With 2 mm input beam: f = f# × D = 3.54 × 2 = 7.08 mm. **Result:** Use a ~7 mm focal length lens to match the focused cone angle to the fiber's acceptance. Couple efficiency depends on mode matching (Gaussian beam waist vs fiber MFD), alignment precision (≤1 µm), and AR coating on fiber end face. Typical efficiency 50–80% in practice.
**Scenario:** A building has legacy OM1 multimode fiber (NA 0.275, 62.5 µm core) at 850 nm. Estimate modal dispersion per km. **Calculation:** V = (π × 62.5 / 0.85) × 0.275 = 63.5. Mode count ≈ V²/2 ≈ 2000. Modal dispersion for step-index: Δτ/L ≈ NA²/(2c × n_core) ≈ (0.275)²/(2 × 3×10⁸ × 1.483) = 8.5 × 10⁻¹¹ s/m = 85 ns/km. Graded-index reduces this ~10× for OM1: ~5 ns/km in real OM1. **Result:** OM1 fiber's ~5 ns/km modal dispersion limits 1 Gbps to ~200 m and 10 Gbps to ~33 m. Modern OM4 reduces to 0.4 ns/km via better refractive index profile, supporting 10 Gbps to 550 m. Single-mode has essentially zero modal dispersion (only chromatic dispersion), enabling 100+ km links.
**Use optical fiber NA calculations for:**
- **Fiber selection**: choosing single-mode vs multimode for specific applications. - **Fiber coupling design**: matching laser sources to fiber acceptance angles. - **Connector and splice design**: ensuring acceptance cones align. - **Bend-loss prediction**: high-NA fibers tolerate tighter bends. - **Multimode fiber characterization**: bandwidth × distance product depends on mode structure. - **Specialty fiber design**: photonic crystal fibers, double-clad fibers for fiber lasers. - **Telecom link planning**: distance, dispersion, attenuation budget. - **Sensor design**: fiber-based gyroscopes, distributed acoustic sensing.
**NA vs application:**
- **Low NA (0.1–0.14)**: telecom single-mode. Good for long distance (small core = single-mode = low dispersion). - **Mid NA (0.2–0.3)**: graded-index multimode, OM3/OM4/OM5. Data center 10G/40G/100G. - **High NA (0.4–0.6)**: large-core multimode, high-power laser delivery. - **Very high NA (>0.6)**: plastic optical fiber, photonic crystal fiber, specialty applications.
**Why low NA for telecom:**
- Smaller core diameter → fewer modes → lower modal dispersion → higher bandwidth × distance. - Single-mode operation requires V < 2.405; achievable with NA ≈ 0.13 and core ~9 µm at 1550 nm. - Coupling is easier with low NA because acceptance cone is narrower (matches small focused spots).
**Bandwidth × distance product:**
| Fiber | BW × distance | |---|---| | OM1 (62.5 µm legacy) | ~200 MHz·km at 850 nm | | OM2 (50 µm graded) | ~500 MHz·km | | OM3 (laser-optimized) | ~2000 MHz·km | | OM4 (enhanced) | ~4700 MHz·km | | OM5 (wideband) | ~28000 MHz·km (WDM) | | Single-mode | >100 GHz·km (chromatic dispersion only) |
Multimode is limited by modal dispersion; single-mode by chromatic and polarization-mode dispersion.
**Fiber attenuation:**
| Wavelength | Telecom SMF loss | |---|---| | 850 nm | ~2 dB/km | | 1310 nm | ~0.35 dB/km | | 1550 nm | ~0.20 dB/km |
Multimode is similar at 850 nm; doesn't work well at 1310/1550 due to high modal dispersion in chromatic-shifted regime.
**Real-world fiber numbers:**
- **OS2 SMF**: 0.18 dB/km at 1550, supports 80+ km links with EDFA amplification. - **OM4 MMF**: 2.5 dB/km at 850, supports ~150 m at 10G, 100 m at 40G/100G. - **PCF (photonic crystal)**: 50 dB/km typical; high NA up to 0.8.
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Core diameter for V-number calculation
NA
0.2425
Acceptance Angle
14.03°
V-Number
24.57
Multimode
| Parameter | Value |
|---|---|
| Numerical Aperture | 0.242487 |
| Acceptance Angle (half) | 14.0334° |
| Acceptance Cone (full) | 28.0668° |
| V-Number | 24.5741 |
| Fiber Type | Multimode |
| Number of Modes | ~302 |
| Relative Index Diff Δ | 1.3422% |
| Core Index | 1.48 |
| Cladding Index | 1.46 |
| Formula | NA = √(n²core - n²clad) |