What is Numerical Aperture?

NA = n × sin(θ) where n is the refractive index of the medium and θ is the half-angle of the maximum cone of light that can enter or exit the optical system. Higher NA means more light-gathering capability AND better diffraction-limited resolution. Used in microscopy, laser optics, and fiber optics.

How are NA and f-number related?

For small angles (paraxial approximation), NA ≈ 1/(2×f/#). More precisely, NA = n × sin(arctan(1/(2×f/#))). The f-number is the ratio of focal length to entrance pupil diameter. Photography uses f#; microscopy uses NA; they're the same quantity in different conventions.

What's the maximum NA in air?

Theoretically 1.0 (when θ = 90°). Practically ~0.95 — beyond that, lens aberrations dominate. Immersion media can push higher: oil immersion at n = 1.52 allows NA up to ~1.4 in practice. Solid immersion lens optics can reach NA > 1.8 with high-index materials.

Why does immersion oil improve microscope resolution?

Resolution d ≈ λ/(2×NA), and NA = n × sin(θ). By filling the gap between sample and objective with oil (n = 1.52), NA can exceed 1.0, giving better resolution. The Abbe limit at NA 1.4 (oil) is ~200 nm — twice as good as NA 0.7 (air). Used for high-magnification, high-resolution imaging.

Why do photographers care about f-number?

F-number controls exposure (lower f# = brighter image, faster shutter possible) and depth of field (lower f# = shallower DOF, blurred background). It's the most-changed setting on most cameras. Stop the lens down (higher f#) for landscape photography (deep DOF); open up (lower f#) for portraits with subject isolation.

How is NA used in fiber optics?

A multimode fiber's acceptance NA determines the maximum angle light can enter and still propagate by total internal reflection. NA = √(n_core² − n_cladding²). For a fiber with NA 0.22, acceptance half-angle is 12.7° in air. Match coupling lens f# to fiber NA for efficient light coupling.

Why is light gathering proportional to NA²?

Solid angle of a cone is approximately π × (half-angle)² for small angles. With NA = sin(θ) ≈ θ for small θ, solid angle ≈ π × NA². Light collected scales with the solid angle, so it goes as NA². Doubling NA quadruples light gathered — the same as halving f-number (two-stop faster lens).

NA to f-Number Converter

Convert between Numerical Aperture and f-number for optical systems. Calculates half-angle, solid angle, and accounts for the refractive index of the medium.

Numerical aperture (NA) and f-number (f-stop) are two ways of describing the same thing: how steeply light cones converge or diverge at a lens. NA is the dominant convention in microscopy and laser optics; f-number is the dominant convention in photography and astronomy. They're related by NA ≈ 1/(2 × f#) for moderate angles, or more precisely NA = n × sin(half-angle of the marginal ray), where n is the refractive index of the medium.

This calculator converts between the two and accounts for the refractive index of the immersion medium. Microscope objectives often use oil (n = 1.515) or water (n = 1.33) instead of air, allowing higher NA values than the air-limited maximum of 1.0. Microscope objectives are typically rated NA 0.1 to 1.4; camera lenses by f# from f/0.95 (very fast) to f/22 (very slow). Both numbers grow inversely: large NA = small f-number = "fast" lens.

Knowing which to use comes down to discipline: - **Microscopy and laser optics**: NA. Higher NA = more light collected, better resolution. - **Photography and cinema**: f-number. Lower f# = brighter, shallower depth of field. - **Astronomy and telescope**: f-ratio (essentially f#). Faster optics for deep-sky; slow optics for high resolution. - **Optical engineering**: both, depending on context.

Inputs

Convert From

Numerical Aperture

f-Number (f/#)

Refractive Index of Medium

1.0 for air, 1.33 for water, 1.52 for oil immersion

Results

0.5000

f-Number

f/0.87

Half-Angle

30.00°

NA / f-Number Results

Parameter	Value
Numerical Aperture	0.5000
f-Number	f/0.87
Half-Angle θ	30.0000°
Solid Angle Ω	0.841787 sr
Medium Index	1.00
Paraxial NA	0.5774
Formula	NA = n × sin(θ)

Last updated: May 29, 2026

Formula

**Numerical aperture definition:** NA = n × sin(θ) Where: - **n**: refractive index of the medium between the object and the lens - **θ**: half-angle of the maximum cone of light that enters/exits the lens **f-number definition:** f# = f / D = 1 / (2 × tan(θ)) Where: - **f**: focal length - **D**: diameter of the entrance pupil - **θ**: half-angle of the marginal ray **Conversion (small-angle / paraxial):** NA ≈ 1 / (2 × f#) (when in air, n = 1) NA ≈ n / (2 × f#) (general, including immersion medium) **Exact conversion:** NA = n × sin(arctan(1 / (2 × f#))) For large NA (small f#), the paraxial approximation breaks down. At NA > 0.5, use the exact formula. **f-number ↔ NA quick reference (air, n = 1):** | f# | NA (paraxial) | NA (exact) | |---|---|---| | f/0.5 | 1.0 | 0.71 | | f/0.7 | 0.71 | 0.58 | | f/1.0 | 0.50 | 0.45 | | f/1.4 | 0.36 | 0.34 | | f/2.0 | 0.25 | 0.24 | | f/2.8 | 0.18 | 0.17 | | f/4.0 | 0.125 | 0.125 | | f/5.6 | 0.089 | 0.089 | | f/8 | 0.063 | 0.063 | | f/16 | 0.031 | 0.031 | | f/22 | 0.023 | 0.023 | At slow speeds (f# > 4), paraxial = exact. **Maximum NA in air:** 1.0 (since sin ≤ 1 and n = 1). **With immersion media (microscopy):** | Medium | n | Max NA | |---|---|---| | Air | 1.0 | 1.0 (theoretical) — practical 0.95 | | Water | 1.33 | 1.33 (practical ~1.2) | | Glycerol | 1.47 | 1.47 (practical ~1.3) | | Standard oil (n=1.515) | 1.515 | 1.515 (practical ~1.4) | | High-index oil (n=1.78) | 1.78 | 1.78 (practical ~1.6) | **Typical NA / f# values by application:** **Microscope objectives:** | Magnification | Typical NA | Equivalent f# | |---|---|---| | 4× | 0.10–0.13 | f/3.8–f/5 | | 10× | 0.25–0.30 | f/1.7–f/2 | | 20× | 0.40–0.50 | f/1.0–f/1.25 | | 40× | 0.65–0.95 | f/0.53–f/0.77 | | 60× (oil) | 1.40 | f/0.36 (n=1.515) | | 100× (oil) | 1.40 | f/0.36 | **Camera lenses:** | Lens type | Typical f# | Equivalent NA | |---|---|---| | Compact zoom | f/3.5–f/5.6 | 0.143–0.089 | | Standard prime | f/1.4–f/2.0 | 0.357–0.250 | | Fast prime | f/0.95 | 0.530 | | Macro lens | f/2.8 | 0.179 | | Sports tele | f/4–f/5.6 | 0.125–0.089 | | Diffraction limit | varies | varies | **Astronomy telescopes:** | Type | Focal ratio | NA | |---|---|---| | Fast Newtonian | f/4–f/5 | 0.125–0.10 | | Schmidt-Cassegrain | f/10 | 0.05 | | Slow refractor | f/15 | 0.033 | | Solar telescope | f/30+ | <0.017 | **Solid angle (steradians):** Ω = 2π × (1 − cos θ) = 2π × (1 − √(1 − NA²/n²)) For NA = 0.5 in air: Ω = 2π × (1 − √0.75) = 2π × 0.134 = 0.84 sr. Solid angle is what determines light collection — important for fluorescence microscopy, astronomical imaging, and any application where you want to capture as much light as possible from a target. **Light gathering vs NA:** Light collected ∝ NA² (small angle) or sin²θ (general). So NA 0.5 collects 4× more light than NA 0.25 — significant for low-light imaging. **Diffraction limit (resolution) vs NA:** For Rayleigh resolution: d ≈ 0.61 × λ / NA For Abbe limit: d ≈ λ / (2 × NA) Higher NA = better resolution AND more light gathering. NA is the single most important spec for a microscope objective.

How to use this calculator

Choose conversion direction (NA → f# or f# → NA).
Enter the NA or f-number value.
Set the refractive index: 1.0 for air, 1.33 for water immersion, 1.52 for oil immersion (most common in microscopy).
Read the converted value plus half-angle and solid angle.
For photography, f# is the standard; for microscopy, NA.
Remember that paraxial approximation breaks at NA > 0.5; use the exact (large-angle) form for high-NA optics.

Worked examples

Photo lens spec interpretation

**Scenario:** A camera lens is rated at f/1.4. What's the equivalent numerical aperture, and what does it mean for light gathering? **Calculation:** NA = 1 / (2 × 1.4) = 0.357 (paraxial). Exact: NA = sin(arctan(1/2.8)) = sin(19.65°) = 0.336. Light gathering: NA²/n² ≈ 0.113 — relative to NA = 0.10 (f/5), this lens gathers ~11× more light per unit time, hence "fast." **Result:** f/1.4 lens has NA ≈ 0.34 in photographic terms. It collects much more light than slow lenses, enabling lower ISO at the same shutter speed. Trade-off: depth of field is shallow at maximum aperture, often used as a portrait aesthetic.

Microscope objective comparison

**Scenario:** Compare a 40× air objective (NA 0.65) with a 60× oil objective (NA 1.4, oil n=1.515). Which gives better resolution? **Calculation:** 40× NA 0.65: d_Abbe = λ/(2×NA) = 550/(2×0.65) = 423 nm. 60× NA 1.4: d_Abbe = 550/(2×1.4) = 196 nm. Light gathering ratio: (1.4)² / (0.65)² ≈ 4.65× more for the oil objective. **Result:** The 60× oil objective gives 2× better resolution (196 vs 423 nm) AND 4.65× more light gathering than the 40× air objective. Oil immersion essentially doubles the effective resolution. Trade-off: short working distance (~0.1 mm), need for oil, and limited to high-magnification work.

Fiber optic acceptance angle

**Scenario:** A multimode fiber with NA = 0.22. What's the equivalent f-number, and what acceptance cone does it have in air? **Calculation:** f# = 1 / (2 × 0.22) = 2.27. Half-angle θ = arcsin(0.22 / 1.0) = 12.7° (in air). Full acceptance cone: 25.4°. **Result:** The fiber accepts light within a 25° cone (12.7° half-angle from axis). Equivalent f# = f/2.3. For coupling: use a focusing lens with f# < 2.3 to ensure the focused beam fills the fiber's acceptance angle. Otherwise, light at angles > 12.7° is lost (the fiber can't propagate it).

When to use this calculator

**Convert NA ↔ f# whenever:**

- **Microscopy spec interpretation**: NA is the standard; many people don't know what NA "means" in f# terms. - **Photography ↔ optical engineering crossover**: photographers think in f#, engineers in NA. - **Fiber optic system design**: NA of the fiber must match the f# of the coupling lens. - **Camera-microscope adapter design**: matching f# of microscope tube lens to camera lens. - **Laser focusing**: NA determines focused spot size; f# determines beam delivery angle. - **Beam expander design**: input/output diameter ratios convert to NA changes.

**When to choose NA vs f#:**

- **NA**: when light gathering and resolution are primary concerns (microscopy, laser focus). - **f#**: when depth of field and exposure time matter (photography). - **Both**: optical engineering, where understanding both conventions is essential.

**Light gathering scales as NA² (or 1/f#²):**

Going from f/4 to f/2 quadruples light gathering — two full stops faster, four times the exposure for same time. Going from NA 0.25 to NA 0.5 also quadruples.

**Depth of field scales as f# (or 1/NA):**

DOF = (n × λ) / NA² (small NA approximation)

Halving NA = doubling DOF. Going from f/2 to f/4 also doubles DOF.

**Practical NA / f# limits:**

- **Maximum in air**: NA = 1.0 (theoretical, impractical); NA = 0.95 (practical microscope objectives). - **Camera lens limits**: f/0.95 (Leica Noctilux); f/0.5 (Zeiss Planar 50/0.7 for NASA, custom). - **Oil immersion microscopy**: NA = 1.4 standard; NA = 1.45 high-end. - **Solid immersion (silicon, n=3.5)**: NA approaches 3.5 — used in IR microscopy.

**Why immersion media work:**

NA = n × sin(θ). For air, n = 1 means max NA = 1 (when θ = 90°). Immersing the gap in oil (n = 1.52) gives max NA = 1.52. This isn't just a number game — the resolution formula uses NA directly (d ≈ λ/(2NA)), so higher NA via immersion really gives better resolution.

**Half-angle vs full cone angle:**

Half-angle θ is from the optical axis to the marginal ray. Full cone angle is 2θ.

For NA = 0.5 in air: half-angle = 30°, full cone = 60°. The light goes from the optical axis out 30° to either side.

**Conversion order matters:**

- **f# → exact NA**: NA = n × sin(arctan(D/(2f))) = n × sin(arctan(1/(2 × f#))) - **NA → exact f#**: f# = 1 / (2 × tan(arcsin(NA/n)))

For small NA, both formulas reduce to NA ≈ n/(2 × f#).

Common mistakes to avoid

Using paraxial approximation at high NA. At NA > 0.5, sin and tan diverge significantly.
Forgetting refractive index of immersion medium. Oil/water immersion lenses have very different NA limits than air.
Confusing NA with focal length. NA describes the cone angle; focal length describes magnification.
Treating NA as a "more is always better." High NA = shallow depth of field, hard alignment, expensive optics.
Mixing photographic and microscopic conventions. Talking about "f/0.5 in microscopy" is correct mathematically but unusual.
Computing NA in terms of D/f instead of sin(θ). The paraxial form is approximate; the sine form is exact.
Forgetting that real-world NA is often less than designed due to aberrations and apodization.

Frequently Asked Questions

Sources & further reading

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