What is the q-parameter?

q(z) = z + i z_R is a complex number that fully describes a Gaussian beam at axial position z relative to the waist. Its real part is the position; its imaginary part is the Rayleigh range. ABCD propagation updates q in one step per element.

Why does the phase advance faster than a plane wave?

The Gouy phase ζ(z) = arctan(z/z_R) adds an extra phase lag near focus. It shifts resonant frequencies in laser cavities and matters in interferometers that compare beams with different focusing histories.

How is the thin lens modeled?

The paraxial thin-lens ABCD matrix [[1, 0], [−1/f, 1]] transforms q at the lens plane; free-space segments before and after use [[1, d], [0, 1]]. The waist is fixed at z = 0; the lens sits at a positive z.

Higher-order modes, aberrations, vector polarization, thermal lensing, and clipping at apertures are not included. The medium is uniform with n = 1.

Gaussian Beam & q-Parameter

A paraxial Gaussian beam is characterized by waist w₀ at z = 0, wavelength λ, and Rayleigh range z_R = πw₀²/λ. The spot size grows as w(z) = w₀√[1 + (z/z_R)²], the wavefront radius of curvature is R(z) = z[1 + (z_R/z)²], and the Gouy phase is ζ(z) = arctan(z/z_R). The complex q-parameter q(z) = z + i z_R encodes both size and curvature; propagation through optical elements uses the ABCD law q₂ = (A q₁ + B)/(C q₁ + D). Free space over distance d has matrix [[1, d], [0, 1]]; a thin lens of focal length f has [[1, 0], [−1/f, 1]]. This simulator plots w(z) and ζ(z) along the axis, applies an optional lens at a user-chosen position, and reports q, w, R, and ζ at an observation plane — the standard toolbox for laser focusing, fiber coupling, and optical trap design.

Who it's for: Undergraduate laser optics and photonics students after thin lenses and before Fourier optics or cavity modes.

Key terms

Gaussian beam
Beam waist
Rayleigh range
Gouy phase
q-parameter
ABCD matrix
Thin lens
Beam propagation

Live graphs

How it works

Gaussian beam propagation: waist w₀, Rayleigh range, Gouy phase, and q-parameter through free space and a thin lens (ABCD matrix).

Key equations

w(z) = w₀ √(1 + (z/z_R)²) · z_R = π w₀² / λ

ζ(z) = atan(z/z_R) · q₂ = (A q₁ + B)/(C q₁ + D)

Frequently asked questions

What is the q-parameter?: q(z) = z + i z_R is a complex number that fully describes a Gaussian beam at axial position z relative to the waist. Its real part is the position; its imaginary part is the Rayleigh range. ABCD propagation updates q in one step per element.
Why does the phase advance faster than a plane wave?: The Gouy phase ζ(z) = arctan(z/z_R) adds an extra phase lag near focus. It shifts resonant frequencies in laser cavities and matters in interferometers that compare beams with different focusing histories.
How is the thin lens modeled?: The paraxial thin-lens ABCD matrix [[1, 0], [−1/f, 1]] transforms q at the lens plane; free-space segments before and after use [[1, d], [0, 1]]. The waist is fixed at z = 0; the lens sits at a positive z.
What is left out?: Higher-order modes, aberrations, vector polarization, thermal lensing, and clipping at apertures are not included. The medium is uniform with n = 1.

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Live graphs

How it works

Key equations

Frequently asked questions

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Frequently asked questions