Rayleigh Range Beam Calculator

Inputs

Beam waist radius w0

nmµmmmcmm

Radius at the focus (1/e² intensity).

Vacuum wavelength λ0

nmµmmmm

Wavelength in vacuum or air (approx.).

M² (beam quality)

Use 1 for an ideal Gaussian beam.

Refractive index n

n=1 for air; >1 for optical materials.

Optional propagation distance z

nmµmmmcmm

Adds w(z), Gouy phase, and intensity ratio.

Output length unit nmµmmmcmm

Controls displayed w0, zR, and related lengths.

Calculate Reset

Formula used

For a Gaussian beam with waist radius w0, vacuum wavelength λ0, refractive index n, and beam quality factor M², the Rayleigh range is:

zR = π × w0² × n ⁄ (M² × λ0)

If n = 1 and M² = 1, this reduces to the ideal Gaussian result.

How to use this calculator

1. Enter the beam waist radius w0 at the focus.
2. Enter the vacuum wavelength λ0 and choose units.
3. Set M² to model real beams; keep 1 for ideal.
4. Set refractive index n for propagation in a medium.
5. Optionally enter distance z to compute w(z) and phase.
6. Press Calculate to view results above the form.

Example data table

Values are rounded for display and depend on selected units.

Rayleigh range beam guide

1) What Rayleigh range represents

Rayleigh range is the distance from the beam waist to the point where the beam area doubles and the radius grows by sqrt(2). It defines the near-focus region where a Gaussian beam stays relatively tight and where phase curvature changes rapidly.

2) Core inputs and units

This calculator uses the waist radius w0 and the vacuum wavelength lambda0, then applies refractive index n to obtain the wavelength in the medium (lambda = lambda0/n). Because w0 is squared, a 10% error in waist measurement produces about a 20% change in Rayleigh range, so waist characterization matters.

3) Beam quality with M^2

Real beams are rarely ideal. The M^2 factor expands divergence and shortens the Rayleigh range compared with a perfect Gaussian. For example, increasing M^2 from 1.0 to 1.5 reduces zR by 33% at the same w0 and lambda0, and it raises the far-field divergence by the same factor.

4) Confocal parameter and depth of focus

The confocal parameter b = 2zR is commonly used as a depth-of-focus metric in scanning, microscopy, and material processing. Within plus or minus zR, spot size stays within 41% of the minimum radius, which helps estimate positioning tolerances for targets and sample stages.

5) Divergence and far-field behavior

Far from the waist, the beam radius grows approximately linearly with distance. The calculator reports the divergence half-angle theta, useful for checking whether an optic will overfill an aperture or whether a beam will clear mechanical constraints over a known path length.

6) Propagation distance option

When you enter a distance z, the tool also estimates w(z), the on-axis intensity ratio I(z)/I(0), Gouy phase, and wavefront curvature. These values support alignment tasks, waist placement checks, and predicting the spot size on a sensor, screen, or workpiece.

7) Typical laboratory ranges

With w0 = 25 um and lambda0 = 1064 nm in air, zR is about 1.84 mm and theta about 13.6 mrad, consistent with common fiber and DPSS laser focusing setups. In glass with n about 1.5, zR increases by roughly 50% for the same w0 and lambda0 because the wavelength shortens in the medium.

8) Practical design tips

Choose w0 based on the required spot size, then confirm zR covers the working distance where beam size must remain stable. If you must extend zR, increase w0, use a longer wavelength, or reduce M^2 with improved mode quality. Always verify units, since mixing nm and um is a frequent error source.

FAQs

1) What is the Rayleigh range in simple terms?

It is the distance from the focus to where the beam radius increases by about 41% (a factor of sqrt(2)). Beyond this point the beam begins to spread more noticeably.

2) Why does the calculator ask for M^2?

M^2 models real, non-ideal beams. Larger M^2 increases divergence and reduces Rayleigh range. Use M^2 = 1 only when the beam is close to a perfect Gaussian mode.

3) How does refractive index affect the result?

In a medium, the wavelength becomes lambda0/n, so the Rayleigh range increases roughly in proportion to n for the same waist and vacuum wavelength. This helps estimate focusing behavior in glass, water, or crystals.

4) What does the confocal parameter mean?

The confocal parameter is 2zR. It represents the full length of the near-focus region around the waist and is often used as a practical depth-of-focus estimate in optical setups.

5) If I enter a propagation distance z, what changes?

The tool adds w(z), beam diameter, intensity ratio, Gouy phase, and curvature at that distance. These values help predict spot size on a target and understand phase evolution.

6) Should I enter beam diameter or radius for w0?

Enter the waist radius, not the diameter. If you only know the diameter at focus, divide it by two before entering the value to keep the Rayleigh range calculation correct.

7) Why are my units important here?

Rayleigh range scales with w0 squared divided by lambda. A unit mistake can change results by factors of 1,000 or more. Double-check whether you entered nm, um, or mm before calculating and exporting.