Gaussian Beam Waist Calculator

Overview: What This Gaussian Beam Waist Calculator Does

This calculator estimates how a laser beam with a Gaussian intensity profile expands as it propagates. From the wavelength, initial beam waist radius, and propagation distance, it returns the beam radius (spot size) at that distance. The same formulas also describe the far-field divergence angle and the Rayleigh range, which are key quantities in laser optics and beam delivery systems.

The tool assumes a fundamental Gaussian mode (TEM₀₀) and reports the 1/e² intensity radius, which is the standard definition of beam waist and beam radius in Gaussian beam theory. If you need a beam diameter, you can simply double the computed radius.

Formulas Used (Gaussian Beam Propagation)

A paraxial Gaussian beam is fully described by its wavelength and its minimum spot size, or waist radius, . As the beam propagates along the axial coordinate , the beam radius evolves according to the standard Gaussian beam propagation law:

where is the Rayleigh range, defined by

In more conventional notation (with all symbols clearly indicated):

• Beam radius at distance z (1/e² intensity radius):
  
• Rayleigh range:
  
• Far-field divergence half-angle (small-angle, paraxial approximation):
  

Here

• is the wavelength of the laser light (in meters in the equations, but the calculator accepts nanometers and converts internally).
• is the beam waist radius at the focus (in meters in the equations, but the calculator expects millimeters and converts to meters).
• is the propagation distance from the beam waist along the optical axis (in meters).

How to Use the Gaussian Beam Waist Calculator

1. Enter the wavelength in nanometers (nm).
   Typical values:
   • 405–450 nm: blue/violet semiconductor lasers
   • 532 nm: frequency-doubled Nd:YAG
   • 632.8 nm: HeNe lasers
   • 800–1,100 nm: Ti:sapphire and many diode lasers
   • 1,550 nm: telecom-band fiber lasers
2. Enter the initial beam waist in millimeters (mm).
   This is the 1/e² radius at the focus or at the narrowest point of the beam. For a collimated beam exiting an optical system, it is often measured with a beam profiler or specified by the manufacturer.
3. Enter the propagation distance z in meters (m).
   This is the distance from the waist plane to the plane where you want to know the beam size (for example, a workpiece, an aperture, or a sensor).
4. Run the calculation. The calculator outputs the beam radius at that distance (in the chosen display units) and may also report the corresponding divergence and Rayleigh range, depending on implementation.

If you need the beam diameter at distance , simply compute

Interpreting the Results

The key output, , is the 1/e² radius of the intensity profile. At radius , the intensity has dropped to of its peak value on axis. This is the standard convention in laser optics and is directly compatible with most Gaussian beam formulas.

In many practical designs you will compare the computed spot size to apertures, detectors, or workpiece features:

• If aperture radius > 2–3 × w(z), essentially all the power passes through.
• If aperture radius ≈ w(z), you lose a significant fraction of power in the aperture edges.
• If aperture radius < w(z), you strongly clip the beam and distort it away from a Gaussian profile.

The divergence angle tells you how quickly the beam expands far from the waist. A larger or shorter wavelength reduces , keeping the beam tighter over longer distances. Conversely, a small waist gives a very small focus but increases divergence.

Worked Example

Consider a classic HeNe laser operating at 632.8 nm with a waist radius of 0.50 mm at the focus. You want to know the spot size 2.0 m away.

1. Inputs:
   • Wavelength:
   • Waist radius:
   • Distance:
2. Convert units for the formulas:
3. Compute Rayleigh range:

4. Compute beam radius at z = 2.0 m:

   The factor under the square root is about 1 + (1.61)² ≈ 3.59, whose square root is ≈ 1.89. So

5. Interpretation:
   • The 1/e² radius at 2 m is about 0.95 mm.
   • The 1/e² diameter is about 1.9 mm.
   • An aperture with radius > 3 mm would comfortably pass nearly all the power at that distance.

Key Quantities at a Glance

Assumptions and Limitations

This calculator is based on standard paraxial Gaussian beam theory. Its predictions are very accurate for many laboratory and industrial laser systems, but there are important assumptions:

• Ideal TEM₀₀ mode: The beam is assumed to be a single transverse mode with a Gaussian intensity profile. Strongly multimode or top-hat beams will not follow these formulas exactly.
• Paraxial approximation: The divergence is assumed to be small and propagation close to the optical axis. Extremely tight focusing or very fast optics (low f-number) can introduce deviations.
• No aberrations or clipping: Lenses and mirrors are treated as ideal, and no apertures clip the beam. Real optical elements with aberrations, misalignment, or dust can distort the beam.
• Monochromatic, continuous-wave behavior: The model uses a single wavelength. Very short pulses may be affected by dispersion, nonlinear effects, or temporal chirp, which are not included.
• Free-space propagation: Effects from waveguides, fibers, or strongly refractive media are not modeled. Use dedicated fiber or waveguide models for those cases.

Whenever measurements deviate from the calculator output, check whether any of these conditions are violated. For safety-critical or high-precision designs, validate with experimental measurements or more advanced optical simulations.

Further Reading and References

The formulas implemented here come from standard Gaussian beam optics, as covered in textbooks such as:

• A. E. Siegman, Lasers (University Science Books)
• O. Svelto, Principles of Lasers

These references provide deeper derivations, extensions to complex beam parameters, and treatment of real optical resonators. For many day-to-day design tasks, however, the simple expressions used by this calculator are sufficient to estimate spot sizes, select lens focal lengths, and size apertures along a Gaussian beam path.