Optics Conversions

Convert between cone acceptance angle, NA, and F/#, and see how they vary across the field.

Relationships (in air, n = 1)

$\text{NA} = \sin\!\left(\theta_{\text{half}}\right)$ where $\theta_{\text{half}}$ is the half-angle of the acceptance cone

$\text{F\#} = \dfrac{1}{2\,\text{NA}}$ (paraxial / image-space in air)

$\theta_{\text{full}} = 2\arcsin(\text{NA})$

Enter any one of the three values and click Calculate. The others are derived automatically.

Half-angle (θ½) °

Numerical Aperture —

F-number (F/#) —

Effective NA and F/# at field angle φ

As the field angle $\varphi$ increases, the marginal ray bundle tilts relative to the optical axis. For a rotationally symmetric lens without vignetting, the projected aperture shrinks by $\cos\varphi$:

$$\text{NA}_\text{eff}(\varphi) = \text{NA}_0 \cdot \cos\varphi$$

$$\text{F\#}_\text{eff}(\varphi) = \frac{\text{F\#}_0}{\cos\varphi}$$

This is the first-order (paraxial) result underlying the $\cos^4\!\varphi$ illumination falloff law. Real systems with vignetting will show steeper rolloff.

On-axis NA —

Max field angle ° (half)

Solid angle of a cone (spherical cap)

$$\Omega = 2\pi\!\left(1 - \cos\theta\right) \quad \text{steradians}$$ where $\theta$ is the half-cone angle (angle from axis to edge of cone)

Full sphere: $\Omega = 4\pi \approx 12.566\,\text{sr}$ Hemisphere ($\theta = 90°$): $\Omega = 2\pi \approx 6.283\,\text{sr}$

Fraction of full sphere: $\Omega / 4\pi$

Half-cone angle (θ) °