01 · Waist to Rayleigh range
At the waist, the complex beam parameter is purely imaginary: .
Playground · research instrument
Gaussian Beam Calculator
Enter wavelength, waist radius and distance. The calculator propagates the complex q parameter to give the ideal 1/e² beam size, Rayleigh range, divergence and phase-front curvature — with the full beam envelope drawn to scale.
Independent research instrument — not claimed as MakerPortal shipped product code. Methods, equations, assumptions, and limitations are disclosed so you can inspect what the page does and does not establish.
Beam inputs
Inputs describe an ideal circular TEM₀₀ beam in free space. Waist is the 1/e² intensity radius, not diameter.
Sign convention: z = 0 at the waist. Positive and negative distances have the same spot size; phase-front curvature changes sign across the waist.
Send the beam through lenses in the optics bench →Ideal free-space envelope
1/e² beam radius versus distance
Target: z = 1.00 m
Vertical scale follows the computed envelope; the optical axis and horizontal distance use separate scales.
Calculated beam
- Rayleigh range zR
- 0.738156 m
- Spot radius w(z)
- 841.915 µm
- Spot diameter
- 1683.830 µm
- Half-angle θ
- 677.363 µrad
- Curvature R(z)
- 1.544875 m
- Depth of focus 2zR
- 1.476312 m
q(z) = 1.000000 + 0.738156i m
What the calculator solves
02 · ABCD propagation
Free space over z uses , so the solver applies the same q propagation used by the full optics bench.
03 · Radius from q
The real part gives phase-front curvature R; the imaginary part gives the 1/e² intensity radius w.
Model boundary. This is scalar, paraxial, monochromatic Gaussian optics with M² = 1. It does not model aperture clipping, elliptical or astigmatic beams, multimode structure, material dispersion, lens thickness or aberrations. For lenses and resonators, continue in the ABCD optics bench.
Two worked examples
Both are solved when the page is built, by the same q-propagation the calculator runs, and checked against the closed-form textbook results in the site's test suite. The same function answers /api/v1/gaussian-beam, so either example can be embedded in your own page or called as JSON.
A · How far a HeNe beam stays collimated
λ = 632.8 nm, waist radius w₀ = 0.5 mm.
- Rayleigh range
- 1.241 m
- Divergence θ = λ/(πw₀)
- 0.403 mrad
- Radius at 10 m
- 4.06 mm
Ten metres is about 8.1 Rayleigh ranges, so the beam is deep in its far field and growing almost linearly with distance.
B · Focusing a 1064 nm beam with a lens
Collimated input radius W = 2 mm, thin lens f = 100 mm.
- Focused waist radius w₀′
- 16.93 µm
- Waist position behind the lens
- 99.9928 mm
- Focused Rayleigh range
- 0.847 mm
- Depth of focus (2 × Rayleigh range)
- 1.69 mm
The waist sits 0.0072 mm inside the geometric focus: a Gaussian beam focuses slightly short of f, by an amount that grows as f approaches the input beam's own Rayleigh range.
Gaussian beam questions
How do I calculate the beam waist after a focusing lens?
For a collimated beam of 1/e² radius W focused by a lens of focal length f, the new waist is w₀′ = W·(f/z_R)/√(1+(f/z_R)²), with z_R = πW²/λ of the incoming beam, which is very close to λf/(πW) whenever f is much shorter than z_R. Worked example: a 1064 nm beam of 2 mm radius through a 100 mm lens focuses to w₀′ = 16.93 µm, 99.9928 mm behind the lens, with a Rayleigh range of 0.847 mm.
What is the Rayleigh range of a laser beam?
The Rayleigh range z_R = πw₀²/λ is the distance from the waist at which the beam's area has doubled; within ±z_R the beam is close to collimated. Worked example: a 632.8 nm HeNe beam with a 0.5 mm waist radius has z_R = 1.241 m, diverges at 0.403 mrad, and has grown to 4.06 mm radius 10 m away.
Is beam waist a radius or a diameter?
The waist w₀ is a radius: the distance from the optical axis where an ideal Gaussian beam’s intensity falls to 1/e² of its on-axis value. The calculator reports both radius and diameter to make that convention explicit.
What happens at one Rayleigh range from the waist?
At z = zR, the beam radius is √2 times the waist radius, the cross-sectional area has doubled, and the phase-front radius of curvature is 2zR.
Does this calculator include M² or lens aberrations?
No. It solves an ideal TEM₀₀ beam with M² = 1 in free space. A measured beam with M² greater than one expands faster, while clipping, astigmatism, chromatic dispersion and lens aberrations require a richer model.
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The instrument, captured—not illustrated.
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