Rayleigh Length Calculator (Gaussian Beam)
Use this tool to calculate the Rayleigh length zR, confocal parameter, and diffraction-limited divergence for a focused Gaussian laser beam.
where w0 is beam waist radius, λ is wavelength, and n is refractive index.
What is Rayleigh length?
The Rayleigh length is the distance from the beam waist to the point where a Gaussian beam’s cross-sectional area doubles (equivalently, where the beam radius increases by a factor of √2). In practical optics, this tells you how long the beam stays tightly focused before diffraction noticeably spreads it.
Rayleigh length formula
For an ideal Gaussian beam, the Rayleigh length is:
- zR: Rayleigh length
- w0: beam waist radius (not diameter)
- λ: wavelength
- n: refractive index of the propagation medium
Important unit reminder
Use consistent units internally (the calculator converts everything to meters first). If you accidentally mix radius and diameter, your result can be off by a factor of 4 because w0 is squared in the formula.
How to use this calculator
- Enter the laser wavelength and choose its unit.
- Enter the beam waist radius and choose its unit.
- Set refractive index n (air is approximately 1.0).
- Choose an output unit and click Calculate.
What results mean
Rayleigh length (zR)
This is your “tight focus zone” on one side of focus.
Confocal parameter (b = 2zR)
This is the full depth of focus around the waist for many practical systems.
Divergence angle
The calculator reports both half-angle and full-angle divergence, useful for estimating spot growth at distance.
Applications in laser optics
- Laser cutting, engraving, and welding process windows
- Microscopy and multiphoton imaging focus control
- Fiber coupling and free-space beam delivery
- Optical trapping and precision metrology
Common mistakes to avoid
- Using beam diameter when the formula requires radius
- Ignoring refractive index when propagating in water, glass, or other media
- Confusing ideal Gaussian behavior with real M2 > 1 beams
Advanced note (real beams)
Real beams often have quality factor M2 greater than 1. A common approximation is: zR,real ≈ zR,ideal/M2. If your measured depth of focus is shorter than expected, beam quality is often the reason.
Quick example
For a 1064 nm laser with a 50 µm waist radius in air (n = 1), the Rayleigh length is about 7.38 mm, so the confocal parameter is about 14.76 mm. That means your highest-intensity focus region is fairly compact and alignment-sensitive.