Optics / 05 / Lens focusing

Where the beam focuses,
and how small.

Send a Gaussian beam through one or two thin lenses. Get the new waist, where it sits, how long it stays focused, and the beam size on each lens.

Ideal thin lenses in air. No aberrations and no clipping: a real lens gives a larger spot when the focus is tight or the beam fills the lens. Runs in your browser; nothing is sent or stored.

Input beam

Vacuum wavelength

Input waist radius W₀ (1/e²)

A collimated laser output: its radius at the output, with the waist there.

Beam quality M²

Distance from the input waist to lens 1

Negative if the beam would reach its waist only after the lens position.

Lenses

Focal length of lens 1

Negative for a diverging lens. Two lenses spaced f₁ + f₂ form a telescope (example: 100 mm and 200 mm, 300 mm apart, enlarge the beam 2×).

Add a second lens (telescope or expander)

Result

New waist radius W₀′: —
Waist position: —
Waist diameter 2W₀′: —
Rayleigh range z_R′: —
Depth of focus 2z_R′: —
Divergence half-angle θ′: —
Waist magnification W₀′/W₀: —
Beam diameter on the lenses: —

Solid vertical lines: the lenses (L1, L2). Blue: ±W, the 1/e² (second-moment) envelope. Marker: the new waist. Distances are measured from the input waist; values are in the table below.

Values in a table

The beam at each lens and the waist that follows it
Lens	Position	f	W on the lens	Waist after it	W₀′

Model and limits

q = z + i z_{R}

complex beam parameter; z measured from the waist

q \to q + d, \frac{1}{q} \to \frac{1}{q} - \frac{1}{f}

free space of length d; thin lens of focal length f

{z_{R}}^{'} = Im q, {W_{0}}^{'} = \sqrt{\frac{M^{2} λ_{0} {z_{R}}^{'}}{π}}

after a lens the waist lies at −Re q behind it (negative: virtual)

{W_{0}}^{'} = \frac{W_{0}}{\sqrt{1 + {(\frac{z_{R}}{f})}^{2}}}, s^{'} = \frac{f}{1 + {(\frac{f}{z_{R}})}^{2}}

one lens with the input waist on it: new waist and its distance

The focus is not at the focal point: the new waist lies at f/(1 + (f/z_R)²) after a lens with the input waist on it, closer to the lens when the input Rayleigh range is short.
M² > 1: the embedded-Gaussian model (Siegman 1990); W is the second-moment radius.
Thin lenses in air, no aberrations: with a fast lens or a tight focus, spherical aberration makes the real spot larger than shown. A round aperture of 1.5 times the 1/e² diameter passes 98.9 % of the power (1 − e^(−4.5)); a smaller one clips the beam and changes the focus.
Paraxial: a note from θ = 0.1 rad and a warning above 0.3 rad anywhere on the path, the levels used in LaserShark.

Sources