Interactive Laser Beam Divergence Calculator
How to Use This Calculator
This calculator determines the divergence angle, Rayleigh range, and spot size at a distance for a laser beam.
- Enter the Wavelength (λ) of the laser light and select its units (nm, µm).
- Enter the Initial Beam Waist Diameter (D0) at the laser output (or beam's narrowest point) and select its units (mm, µm). This is 2w0.
- Enter the Beam Quality Factor (M²). Use 1 for an ideal TEM00 Gaussian beam. Real beams have M² > 1.
- Optionally, enter the Distance from Laser (z) where you want to calculate the spot size, and select its units (m, cm, mm).
- Click the "Calculate Divergence & Spot Size" button.
- Results and a step-by-step solution will be displayed.
Theory: Laser Beam Divergence
Laser beams, even if perfectly collimated, naturally spread out (diverge) as they propagate due to diffraction. This spreading is characterized by the beam divergence angle (θ).
Gaussian Beams: The ideal laser beam profile is a Gaussian beam (TEM00 mode). Its intensity is highest at the center and decreases radially. The beam waist (w0) is the radius of the beam at its narrowest point. The diameter D0 = 2w0.
Divergence Angle (θ): This is the half-angle at which the beam radius or diameter increases with distance in the far field. For a Gaussian beam, it's given by θ ≈ λ / (πw0).
Rayleigh Range (zR): This is the distance from the beam waist to the point where the cross-sectional area of the beam doubles. Within this range (from -zR to +zR), the beam is approximately collimated. $z_R = \pi w_0^2 / \lambda$.
Beam Quality Factor (M²): Real laser beams are not perfectly Gaussian. The M² ("M-squared") factor quantifies how much a real beam's divergence deviates from an ideal Gaussian beam with the same waist size. M² = 1 for an ideal beam; M² > 1 for real beams. The divergence of a real beam is M² times larger than that of an ideal beam: θreal = M² × θideal. The Rayleigh range for a real beam is $z_R = \pi w_0^2 / (M^2 \lambda)$.
Spot Size at Distance z: The beam radius w(z) at a distance z from the waist is $w(z) = w_0 \sqrt{1 + (z/z_R)^2}$. The diameter D(z) = 2w(z).
Formulas Used
- 1. Beam Waist Radius:
w0 = D0 / 2 - 2. Divergence Half-Angle (θ):
θ = M²λ / (πw0)(in radians) - 3. Rayleigh Range (zR):
zR = πw0² / (M²λ) - 4. Beam Radius at distance z, w(z):
w(z) = w0 √(1 + (z/zR)²) - 5. Beam Diameter at distance z, D(z):
D(z) = 2w(z)