NA = n · sin(θ).Formula Used
Numerical aperture measures the light‑gathering ability of an optical system and the acceptance cone of a fiber or objective.
- Main definition:
NA = n · sin(θ), wherenis the medium refractive index andθis the cone half‑angle. - Angle from NA:
θ = arcsin(NA / n)(real only ifNA ≤ n). - Paraxial relation:
F# ≈ n / (2·NA), useful for quick comparisons. - Lateral resolution (Rayleigh):
d ≈ 0.61·λ / NA. - Depth of focus (approx):
DOF ≈ 2·n·λ / NA². - Collection solid angle:
Ω = 2π(1 − cosθ).
How to Use This Calculator
- Select a Solve for mode based on what you know.
- Choose degrees or radians for the half‑angle input.
- Enter the required values. Use realistic ranges (e.g.,
0 ≤ NA ≤ n). - Click Calculate. Results appear above the form.
- Use Download CSV or Download PDF after a calculation.
Tip: For microscopy, keep wavelength and medium consistent when comparing objectives.
Example Data Table
| Medium n | Half-angle θ (deg) | NA = n·sin(θ) | λ (nm) | Resolution d (nm) |
|---|---|---|---|---|
| 1.000 | 30 | 0.500 | 550 | 671 |
| 1.33 | 40 | 0.855 | 550 | 392 |
| 1.52 | 60 | 1.316 | 550 | 255 |
Values are illustrative and assume the Rayleigh approximation.
Understanding Numerical Aperture in Practice
Numerical aperture (NA) summarizes how strongly an optical system accepts or delivers light. In microscopy, higher NA usually means brighter images and finer detail. In fiber optics, NA describes the acceptance cone for coupling light into the core.
Key Definition and Physical Meaning
The core relationship is NA = n·sin(θ), where n is the surrounding medium index and θ is the half-angle of the cone. A larger θ increases collection solid angle, improving photon capture for weak signals.
Typical NA Ranges You Will See
Air objectives often range from about 0.10 to 0.95. Water immersion commonly reaches 1.0 to 1.2, while oil immersion objectives may reach 1.3 to 1.4. The condition NA ≤ n must hold for a real cone angle.
Resolution Estimates Using Wavelength
This calculator reports a Rayleigh-style lateral resolution, d ≈ 0.61·λ/NA. For λ = 550 nm and NA = 0.50, d is about 671 nm. At NA = 1.30, d improves to roughly 258 nm, highlighting why high-NA objectives reveal finer features.
Depth of Focus and Imaging Tolerance
Higher NA can reduce depth of focus. A useful approximation is DOF ≈ 2·n·λ/NA². For λ = 550 nm in air (n = 1.0), DOF is about 4.4 µm at NA = 0.50, but near 0.65 µm at NA = 1.30.
Link Between NA and F-number
For paraxial optics, NA relates to f-number by F# ≈ n/(2·NA). This is an approximation that becomes weaker at large angles, but it is practical for fast comparisons. For example, in air, NA = 0.25 corresponds to roughly F/2.
Fiber Acceptance Cone and Coupling
In step-index fibers, a larger NA generally eases alignment and improves coupling from LEDs or multimode sources. However, higher NA can support more modes and may increase modal dispersion, which matters in long-distance links and high-bandwidth systems.
How to Use Results for Design Decisions
Use NA to compare objectives, estimate required illumination, and check whether angles are physically possible in a given medium. Keep λ consistent when comparing resolution. If you switch immersion media, update n to reflect the real environment near the lens or fiber.
For photon-limited imaging, NA can strongly influence signal-to-noise because it affects both excitation concentration and detection efficiency. The reported solid angle helps approximate collection fraction for isotropic emitters. When documenting a setup, record the medium, wavelength band, and any stops that limit the pupil.
FAQs
1) What does a higher numerical aperture mean?
A higher NA usually means a wider acceptance cone, more collected light, and better lateral resolution. It can also reduce depth of focus, making focus and sample flatness more critical.
2) Can NA be greater than 1?
Yes, when the medium index is above 1. Oil or water immersion objectives can have NA values above 1. The practical limit for real angles is NA ≤ n of the surrounding medium.
3) Which angle should I enter as θ?
θ is the half-angle of the light cone in the medium. If you know the full cone angle, divide it by two. For fibers, θ relates to the acceptance cone around the fiber axis.
4) How accurate is the f-number mode?
The relation NA ≈ n/(2·F#) is a paraxial approximation. It is useful for small angles and quick comparisons, but real systems can deviate due to pupil shape, vignetting, and design details.
5) Why does wavelength change the resolution result?
Diffraction scales with wavelength. Shorter wavelengths typically yield smaller diffraction-limited spots for the same NA. That is why the calculator requests λ for resolution and depth estimates.
6) Why does immersion increase effective performance?
Immersion media raise the refractive index near the lens, allowing larger NA and a wider cone of collected light. This can improve brightness and resolution compared with air for the same geometry.
7) Is NA enough to compare two objectives?
NA is a strong predictor of brightness and resolution, but not the only factor. Field flatness, aberration correction, transmission, working distance, and coating quality also strongly affect real image performance.