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.
**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.
**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.
**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).
**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#).
Calculate optical fiber numerical aperture, acceptance angle, and V-number. NA = sqrt(n_core² - n_clad²).
Calculate image distance, magnification, and image characteristics using the thin lens equation 1/f = 1/do + 1/di.
Calculate focal length from magnification, object/image distance, or field of view. Versatile focal length solver.
Design a grating pulse compressor: calculate grating separation, compressed pulse duration, GDD, and chirp parameter.
Calculate pulse front tilt angle and group delay from grating parameters for ultrafast optics.
Calculate focal length and optical power using the Lensmaker's equation: 1/f = (n-1)[1/R1 - 1/R2].
1.0 for air, 1.33 for water, 1.52 for oil immersion
NA
0.5000
f-Number
f/0.87
Half-Angle
30.00°
| 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) |