What is the V-number?

The V-number (normalized frequency) V = (π × d × NA) / λ determines the number of modes a fiber can support. When V < 2.405, only the fundamental mode propagates (single-mode fiber). For higher V, multiple modes propagate; mode count ≈ V²/2 for step-index, V²/4 for graded-index.

What determines single-mode vs multimode?

A fiber is single-mode when V < 2.405, where V = πdNA/λ. Single-mode fibers have small core diameters (typically 8-10 μm) while multimode fibers have larger cores (50-62.5 μm). Single-mode supports much higher bandwidth × distance because modal dispersion is essentially zero.

What is the typical NA of telecom single-mode fiber?

About 0.13–0.14 for standard SMF-28 (Corning), giving an acceptance half-angle of ~8°. This small NA enables single-mode operation at 1310/1550 nm with reasonable core size (8.2 µm). Low NA also reduces fiber-to-fiber coupling losses.

How is fiber NA different from microscope NA?

Both describe a cone of light acceptance via NA = n × sin(θ). For fibers, NA derives from core/cladding indices: NA = √(n_core² − n_clad²). For microscopes, NA is the cone collected by the objective from the sample (depends on lens design and immersion medium).

What is modal dispersion?

In multimode fibers, different modes travel at different velocities, causing pulses to spread. Step-index fibers have severe modal dispersion (~50 ns/km); graded-index reduces this to ~5 ns/km. Single-mode fibers have only one mode, so modal dispersion is essentially zero (only chromatic and PMD remain).

Why is single-mode fiber preferred for long distances?

Single-mode has near-zero modal dispersion, supporting >100 GHz·km bandwidth-distance product (vs ~5 GHz·km for OM4 multimode). Single-mode also has lower attenuation (0.2 dB/km at 1550 nm vs ~2 dB/km for multimode at 850 nm) and works at the EDFA gain band, enabling amplification for very long links.

How do I couple light into a fiber efficiently?

Match the focused beam to the fiber's mode field diameter (MFD), not just core diameter. Ensure the focused cone angle is within the acceptance angle (sin θ ≤ NA). Use a coupling lens with appropriate f-number, anti-reflection-coated fiber end face, and precise XYZ alignment. Typical coupling efficiency 50–80%; commercial pigtailed lasers achieve >90%.

Optical Fiber NA Calculator

Q: What determines single-mode vs multimode?

A fiber is single-mode when V < 2.405, where V = πdNA/λ. Single-mode fibers have small core diameters (typically 8-10 μm) while multimode fibers have larger cores (50-62.5 μm). Single-mode supports much higher bandwidth × distance because modal dispersion is essentially zero.

Q: What is the typical NA of telecom single-mode fiber?

About 0.13–0.14 for standard SMF-28 (Corning), giving an acceptance half-angle of ~8°. This small NA enables single-mode operation at 1310/1550 nm with reasonable core size (8.2 µm). Low NA also reduces fiber-to-fiber coupling losses.

Q: How is fiber NA different from microscope NA?

Both describe a cone of light acceptance via NA = n × sin(θ). For fibers, NA derives from core/cladding indices: NA = √(n_core² − n_clad²). For microscopes, NA is the cone collected by the objective from the sample (depends on lens design and immersion medium).

Q: What is modal dispersion?

In multimode fibers, different modes travel at different velocities, causing pulses to spread. Step-index fibers have severe modal dispersion (~50 ns/km); graded-index reduces this to ~5 ns/km. Single-mode fibers have only one mode, so modal dispersion is essentially zero (only chromatic and PMD remain).

Q: Why is single-mode fiber preferred for long distances?

Single-mode has near-zero modal dispersion, supporting >100 GHz·km bandwidth-distance product (vs ~5 GHz·km for OM4 multimode). Single-mode also has lower attenuation (0.2 dB/km at 1550 nm vs ~2 dB/km for multimode at 850 nm) and works at the EDFA gain band, enabling amplification for very long links.

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.

Inputs

Core Refractive Index

Cladding Refractive Index

Core Diameter (μm)

Core diameter for V-number calculation

Wavelength (nm)

Results

0.2425

Acceptance Angle

14.03°

V-Number

24.57

Multimode

Optical Fiber NA Results

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)

Last updated: May 29, 2026

Formula

**Numerical aperture (step-index fiber):** NA = √(n_core² − n_cladding²) Where: - **n_core**: refractive index of the fiber core - **n_cladding**: refractive index of the surrounding cladding **Acceptance angle (in external medium of index n_ext):** θ_acc = arcsin(NA / n_ext) In air (n_ext = 1.0): θ_acc = arcsin(NA). For typical telecom fiber NA = 0.14: θ_acc ≈ 8°. For multimode NA = 0.22: θ_acc ≈ 12.7°. **V-number (normalized frequency):** V = (π × d / λ) × NA = (2π × a / λ) × NA Where: - **d**: core diameter - **a**: core radius (= d/2) - **λ**: free-space wavelength V = 2.405 is the single-mode cutoff. Below this, only the fundamental LP01 mode propagates. **Number of modes (approximate, for step-index multimode):** N ≈ V² / 2 For V = 30 (typical 50 µm multimode at 850 nm): N ≈ 450 modes. For graded-index fiber, divide by 2: N_GI ≈ V² / 4. **Single-mode vs multimode boundaries:** - **V < 2.405**: single-mode (fundamental mode only). - **V = 2.405–3.832**: 2-mode operation. - **V > 3.832**: multimode (rapidly increasing mode count). **Common fiber types:** | Fiber type | Core (µm) | Clad (µm) | NA | Use | |---|---|---|---|---| | SMF-28 (single-mode) | 8.2 | 125 | 0.14 | Telecom long-haul, 1310/1550 nm | | OS2 (low-water peak SMF) | 9 | 125 | 0.13 | 1310–1625 nm coverage | | OM1 (multimode) | 62.5 | 125 | 0.275 | Legacy, 850/1300 nm | | OM2 (multimode) | 50 | 125 | 0.20 | Mid-range data | | OM3, OM4 (laser-optimized) | 50 | 125 | 0.20 | 10G/40G/100G data, 850 nm | | OM5 (wideband) | 50 | 125 | 0.21 | SWDM, 850–953 nm | | Photonic crystal fiber | varies | (air cladding) | varies (often 0.8) | High-NA, single-mode broadband | | Plastic optical fiber (POF) | 1000 | 1500 | 0.5 | Short-distance consumer | **Worked example: standard SMF-28 at 1550 nm** n_core ≈ 1.4682, n_clad ≈ 1.4628 (Δ ≈ 0.36%). Core d = 8.2 µm. NA = √(1.4682² − 1.4628²) = √(2.155 − 2.140) = √0.0157 = **0.125** θ_acc = arcsin(0.125) = **7.2°** V = (π × 8.2 / 1.55) × 0.125 = 16.6 × 0.125 = **2.08** → single-mode at 1550 nm ✓ At 1310 nm: V = (π × 8.2 / 1.31) × 0.125 = 19.6 × 0.125 = 2.46 → just above cutoff; technically two-mode but the LP11 mode is weakly guided, effectively single-mode for telecom purposes. **Worked example: OM4 multimode at 850 nm** n_core ≈ 1.483, n_clad ≈ 1.469 (Δ ≈ 1%). Core d = 50 µm. NA = √(1.483² − 1.469²) = √(2.199 − 2.158) = √0.041 = **0.20** θ_acc = arcsin(0.20) = **11.5°** V = (π × 50 / 0.85) × 0.20 = 184.7 × 0.20 = **36.9** Number of modes ≈ V²/2 = 681 modes (highly multimode). Modal dispersion limits this fiber to ~400 m at 10 Gbps with VCSEL-laser launches. **Coupling efficiency considerations:** To couple light efficiently into a fiber: 1. Focus the beam to match the core diameter. 2. Ensure the focused cone angle is within the fiber's acceptance angle (sin θ_focus ≤ NA). 3. Match modes: for single-mode, the input beam should be Gaussian-shaped with waist ≈ mode field diameter (MFD). Single-mode fiber MFD ≠ core diameter exactly; for SMF-28 at 1550 nm, MFD ≈ 10.4 µm (vs core 8.2 µm).

How to use this calculator

Enter core and cladding refractive indices. For silica fibers: core ~1.46–1.48, cladding ~1.45–1.46 (small differences are key).
Enter the core diameter (8–10 µm for single-mode, 50–62.5 µm for multimode).
Enter the operating wavelength (850, 1310, or 1550 nm for telecom).
Read NA, acceptance angle, V-number, and mode count.
For single-mode operation, ensure V < 2.405 at your wavelength.
For coupling design, match focused beam angle to acceptance angle, and beam waist to mode field diameter.

Worked examples

Designing a fiber for 850 nm 100 Gbps link

**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.

Fiber-coupled laser diode

**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.

Estimating modal dispersion in legacy fiber

**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.

When to use this calculator

**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.

Common mistakes to avoid

Confusing fiber NA with microscope/photography NA. Same formula but the underlying geometry differs.
Forgetting that V depends on operating wavelength. A fiber single-mode at 1550 nm may be multimode at 1310 nm if V crosses 2.405.
Treating "8 µm core" and "10 µm mode field diameter" as the same. MFD is the actual mode size; core diameter is the physical structure.
Computing NA from core/cladding indices without accounting for refractive index profile (step vs graded). Step-index NA formula doesn't apply directly to graded-index fibers.
Forgetting acceptance angle changes with external medium. Underwater, in oil, or in immersion, acceptance angle reduces by n.
Using paraxial coupling formulas at high NA. Real coupling efficiency depends on mode overlap, not just angles.
Ignoring polarization-mode dispersion in single-mode fibers. For very long links, PMD limits practical bandwidth.

Frequently Asked Questions

Sources & further reading

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