How this calculation works
A fundamental Gaussian beam narrows to a waist and then expands with distance. Its radius is defined by an intensity threshold rather than a hard edge. This page propagates an ideal beam and integrates the fraction passing through a centred circular aperture. The Rayleigh range marks where the beam radius becomes √2 times the waist radius. It is not the distance at which the beam suddenly stops being narrow; width changes continuously on both sides of the waist.
zR = πw0²/λ; w(z) = w0√[1+(z/zR)²]; enclosed power = P[1−exp(−2a²/w²)]A worked example
A 0.5 mm waist at 500 nm gives a Rayleigh range of 1570.796 mm. At 1000 mm from that waist, beam radius is about 0.592724 mm. At the waist itself, a centred 0.5 mm radius aperture passes 1 − exp(−2), or about 0.864665 of total power. This provides a simple independent check on the aperture integration.
Good to know
The beam is ideal, fundamental, circular and propagates in free space. Real M², astigmatism, clipping alignment and thermal lensing are absent. The aperture result counts power at that plane before subsequent diffraction. It does not propagate the clipped beam or provide a laser-safety assessment. Use measured beam parameters when applying the model to equipment.
Reference: Edmund Optics — Gaussian beam propagation (opens in a new tab) ↗
What this calculation answers
A fundamental Gaussian beam narrows to a waist and then expands with distance. Its radius is defined by an intensity threshold rather than a hard edge. This page propagates an ideal beam and integrates the fraction passing through a centred circular aperture.
The Rayleigh range marks where the beam radius becomes √2 times the waist radius. It is not the distance at which the beam suddenly stops being narrow; width changes continuously on both sides of the waist.
The Step index fiber numerical aperture and V number calculator derives a geometric acceptance cone and normalized frequency. A Gaussian waist radius is not automatically the same as a fiber core radius.
Use the right input conventions
Waist and aperture entries are radii, in millimetres. The beam radius uses the 1/e² intensity convention. Enter signed axial distance from the waist, so equal positive and negative distances give equal ideal widths.
Wavelength is the free-space value for this M² = 1 model. Inputs that imply a far-field half divergence above 0.1 radians are rejected because the paraxial Gaussian approximation is no longer consistent.
Interpret the result
A Gaussian profile has tails beyond every finite radius. The aperture output integrates that profile instead of assuming all power inside the quoted beam radius and none outside. At aperture radius equal to beam radius, the enclosed fraction is about 86.47 percent.
Peak intensity is twice total power divided by πw² under the stated convention. It is a mathematical irradiance value, not an exposure limit, detector saturation threshold or safe viewing criterion.
Work through the example
A 0.5 mm waist at 500 nm gives a Rayleigh range of 1570.796 mm. At 1000 mm from that waist, beam radius is about 0.592724 mm.
At the waist itself, a centred 0.5 mm radius aperture passes 1 − exp(−2), or about 0.864665 of total power. This provides a simple independent check on the aperture integration.
Where the model stops
The beam is ideal, fundamental, circular and propagates in free space. Real M², astigmatism, clipping alignment and thermal lensing are absent.
The aperture result counts power at that plane before subsequent diffraction. It does not propagate the clipped beam or provide a laser-safety assessment. Use measured beam parameters when applying the model to equipment.
A few questions, answered
Is waist diameter the same input as waist radius?
No. Entering diameter as radius makes Rayleigh range four times too large in this model.
What changes in the comparison?
Move the aperture to the waist and set its radius equal to the waist radius.