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Gaussian Beams and Beam Transformation

Follow a paraxial TEM₀₀ laser beam through up to three thin lenses. The beam radius w(z), wavefront curvature 1/R(z), Gouy phase and complex parameter q are propagated with ABCD matrices, and there are solvers for mode matching and focusing to a spot.

Drag lenses, the input waist or the z cursor on the envelope plot • Arrow keys move the cursor when the plot has focus • Every slider has a number box
zR — Rayleigh range (input)
w — w(z) at cursor
R — R(z) at cursor
ψ — Gouy phase at cursor
w₀′ — Final waist
θ — Max divergence

Beam envelope ±w(z) paraxial

Filled band: ±w(z), the 1/e² intensity radius (vertical scale greatly magnified relative to z). Shaded strips: |z − z₀| < zR for each waist. ◆ waist markers (the input waist is draggable), lenses drawn as ↕ (converging) or ↔-tipped (diverging) and draggable, white line: z cursor. Arcs: surfaces of constant phase, sag z = −r²/(2R) exaggerated by the factor printed on the plot; a straight line means a planar wavefront.

Wavefront curvature 1/R(z) and accumulated Gouy phase ψ(z)

Top: curvature 1/R (m⁻¹, positive = diverging). It passes through exactly 0 at every waist, where R = ∞ and the wavefront is planar; plotting 1/R avoids a false finite value. Bottom: on-axis Gouy phase accumulated from z = 0 (continuous across thin lenses).

Transverse intensity at the cursor

I(x, y) in W/cm² (unit prefix chosen automatically), absolute and power-normalised so that ∫I dA = P. Dashed circle: r = w.

Transverse phase at the cursor

Field phase relative to the plane-wave carrier exp(ikz), φ = k r²/(2R) − ψ, wrapped to (−π, π]; masked (dark) where I < 10⁻³ I₀. Uniform colour means a planar wavefront.

Line cut through the axis

Left: I(x, 0) with ±w markers and the I₀/e² level (the definition of w). Right: unwrapped phase φ(x) = k x²/(2R) − ψ in rad; flat at a waist.

Cursor z
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Beam radius w(z)
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Radius of curvature R(z)
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Gouy phase ψ (accum. / local)
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q = (z − z₀) + i zR
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Local waist w₀ @ z₀
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Local zR, θ
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Peak intensity I₀ = 2P/(πw²)
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∫ I dA / P (Simpson, r ≤ 6w)
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Paraxial check: max θ
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Waist-to-waist transformation at each lens: q-law result, Self's formula, and a direct ABCD check
Lens z (mm) f (mm) s: waist → lens (mm) Output waist w₀′ s″: lens → waist (mm) m = w₀′/w₀ (Self) |Δ| Self vs q-law |Δw| ABCD vs segment
💡 How to Use

Learn with this tool

Learning objectives

  • Relate w₀, λ₀ and n to the Rayleigh range, divergence and depth of focus, and read w, R and ψ off a propagating beam.
  • Use the complex beam parameter q and ABCD matrices to transform a beam through lenses, and explain why a focused Gaussian waist is not at the geometric image.
  • Design a focusing or mode-matching system and judge when the paraxial Gaussian model stops being trustworthy.

Prerequisites

Model

The field is a scalar, monochromatic paraxial beam in a homogeneous medium of index n. With the site convention E = Re{u exp[i(kz − ωt)]}, k = 2πn/λ₀, the fundamental Gaussian mode is

u(r, z) = (w₀/w) exp(−r²/w²) exp[i(k r²/(2R) − ψ)]

zR = π n w₀²/λ₀,  w(z) = w₀√(1 + (z/zR)²),  R(z) = z + zR²/z,  ψ(z) = atan(z/zR)

q = z + i zR,  1/q = 1/R − i λ₀/(π n w²),  qout = (A q + B)/(C q + D)

I(r) = (2P/πw²) exp(−2r²/w²),  ∫ I dA = P

Here z is measured from the waist. The field in the site convention is u = (q₀/q)* exp[ik r²/(2q*)]. The tool uses the Kogelnik–Li bookkeeping form 1/q = 1/R − iλ₀/(πnw²) and q = z + i zR. Every ABCD matrix is real, so q and q* transform identically. Ray matrices act on [y, θ] with θ the geometric angle, as in the ray-optics bench: translation [[1, d], [0, 1]] and thin lens [[1, 0], [−1/f, 1]]. Assumptions: paraxial (θ ≪ 1), thin aberration-free lenses of unlimited aperture, no absorption, and a circularly symmetric TEM₀₀ mode. M² extension: a real beam with M² > 1 is modelled by its second-moment width W, which follows the Gaussian law with λ₀ replaced by M²λ₀. This embedded-Gaussian rule predicts widths only; it does not give the real field or its phase.

w, w₀
1/e² intensity radius (1/e field radius), and its minimum (waist)
zR
Rayleigh range: distance from the waist where w = √2 w₀ and the area doubles
R
wavefront radius of curvature (R > 0 diverging); R = ∞ (planar) at a waist
ψ
Gouy phase: extra on-axis phase lag relative to a plane wave, total π through a focus
q
complex beam parameter, which holds both w and R
θ
far-field half-angle divergence M²λ₀/(π n w₀), in the medium
λ₀, n
vacuum wavelength and refractive index (medium wavelength λ₀/n)
Derivation: from the paraxial wave equation to the q law

Substituting E = u(x, y, z) exp[i(kz − ωt)] into the Helmholtz equation and dropping ∂²u/∂z² (slowly varying envelope) gives the paraxial wave equation ∇⊥²u + 2ik ∂u/∂z = 0. Try u = A(z) exp[ik r²/(2Q)]. The r² terms cancel only if dQ/dz = 1, so Q = z + Q₀: free space adds distance to Q. The r⁰ terms give A ∝ 1/Q. For the field to stay finite as r → ∞, Im(1/Q) > 0. Take Q = q* with q = z + i zR: then 1/q* = 1/R + i λ₀/(π n w²), and the modulus and argument give w(z), R(z) and the Gouy phase ψ = arg(i zR/q) = atan(z/zR).

A thin lens multiplies the field by exp[−ik r²/(2f)], so 1/q* → 1/q* − 1/f. This is exactly q′ = (Aq + B)/(Cq + D) with A = 1, B = 0, C = −1/f, D = 1. Free space gives q′ = q + d, which is the same law with [[1, d], [0, 1]]. Any cascade of these operations is therefore described by the product matrix. The ray matrix of an optical system is also its beam matrix, which is why paraxial ray optics and Gaussian beam optics share one ABCD formalism. Power is conserved because the lens and free-space operators are unitary; in the normalised form this is ∫(2P/πw²)exp(−2r²/w²) 2πr dr = P for every w.

Imaging a waist at distance s before a lens gives Self's lens formula 1/(s + zR²/(s − f)) + 1/s″ = 1/f, with magnification m = 1/√((1 − s/f)² + (zR/f)²). It reduces to the thin-lens equation as zR → 0.

Exercise 1: one Rayleigh range

  1. A 632.8 nm HeNe beam has w₀ = 0.50 mm in air. Predict zR, w(zR) and R(zR).
  2. Load HeNe zR. The cursor sits at z₀ + zR.
  3. Read w and R. Move the cursor either way: is |R| smaller anywhere else? What is R at the waist?
  4. Explain why the wavefront is flattest both at the waist and far away, and most curved at z = zR.
Show answer

zR = π(0.5 mm)²/632.8 nm = 1.241 m, w = √2 w₀ = 0.707 mm and R = 2zR = 2.48 m. R(z) = z + zR²/z has its minimum at z = zR. At the waist the wavefront is planar (R = ∞, 1/R = 0). In the far field R → z, like a spherical wave from the waist.

Exercise 2: a focused waist is not at the geometric image

  1. A 100 µm waist (1064 nm) sits 150 mm before an f = 100 mm lens. Geometric optics puts the image at 300 mm with magnification 2. Predict whether the Gaussian waist lies closer or farther.
  2. Load Waist imaging.
  3. Read s″ and m in the lens table (Self's formula and the q law both appear).
  4. Use zR = 29.5 mm to explain the shift. How must w₀ change to recover the geometric answer?
Show answer

s″ = f + f²(s − f)/((s − f)² + zR²) = 248 mm, and m = 1.72 (w₀′ = 172 µm). This is closer than 300 mm and smaller than ×2. The input is not a point source: its wavefront at the lens has R = s + zR²/s, larger than s. Increasing w₀ makes zR larger and the discrepancy worse; making zR ≪ |s − f| (smaller w₀) recovers geometric imaging.

Exercise 3 (limiting case): waist in the front focal plane

  1. Put the input waist exactly at s = f before a lens. Geometric optics sends the image to infinity. Predict where the Gaussian output waist is and how large it is, for any zR.
  2. Use one lens with f = 100 mm at z₁ = 100 mm and z₀ = 0. Try w₀ = 50 µm, 200 µm and 1 mm.
  3. Read s″ and w₀′ from the table, then compare w₀′ with λ₀f/(πw₀).
  4. Why is the output always in the back focal plane? Relate w₀′ ∝ 1/w₀ to a Fourier transform.
Show answer

With s = f, Self's formula gives s″ = f exactly and m = f/zR, so w₀′ = λ₀f/(πnw₀) for every zR. The lens maps the front focal plane to the back focal plane by an exact Fourier transform, and a Gaussian of width w₀ transforms to a Gaussian of width ∝ 1/w₀. Unlike geometric optics, there is no singularity.

Exercise 4: Gouy phase through a focus

  1. How much extra on-axis phase does a beam gain passing from far before to far after a tight focus?
  2. Load Focus f = 100 and lengthen the bench to 3000 mm.
  3. Read ψ just after the lens and at the end of the bench on the Gouy plot. How wide (in z) is the region where most of the change happens?
  4. Connect this width to zR′ and to the spread of transverse wavevectors in a small spot.
Show answer

The change approaches π (−π/2 → +π/2 about the focus). Half of it happens within ±zR′ ≈ ±3.4 mm of the waist. A narrow spot contains transverse wavevectors k⊥ ∼ 2/w₀′, and their slightly smaller kz gives an average phase lag. This lag sets the frequency spacing of transverse modes in the Fabry–Pérot and laser cavity tools.

Worked example: focusing a Nd:YAG beam

A collimated 1064 nm beam with w = 1.00 mm at its waist hits an f = 100 mm lens placed at that waist (air, P = 1 W). Input: zR = π(1 mm)²/1.064 µm = 2.953 m, so f/zR = 0.0339. q law: qin = i zR, and qout = qin/(1 − qin/f) gives the waist at d = f/(1 + (f/zR)²) = 99.885 mm, which is 0.115 mm before the focal plane. Its radius is w₀′ = w (f/zR)/√(1 + (f/zR)²) = 33.85 µm; the estimate λf/(πw) = 33.87 µm is 0.06 % high. Focal region: zR′ = πw₀′²/λ = 3.38 mm (depth of focus 2zR′ = 6.8 mm), with divergence θ = λ/(πw₀′) = 10.0 mrad, which is well inside the paraxial range. Peak intensity: I₀ = 2P/(πw₀′²) = 5.56 × 10⁴ W/cm². Check with the Focus f = 100 preset (P = 1000 mW) and the readouts.

When the model fails

  • Non-paraxial focusing: when θ = λ₀/(πnw₀) exceeds about 0.1–0.3 rad (w₀ of order λ), the dropped ∂²u/∂z² term matters. Vector (polarisation) effects, spot elongation and a shifted focus then require vectorial diffraction theory (Richards–Wolf).
  • Apertures and aberrations: real lenses clip the beam: an aperture of radius a passes 1 − exp(−2a²/w²) of the power, so the common “diameter = πw” rule loses 0.7 % and a = 3w loses 1.5 × 10⁻⁸ and add spherical aberration, which the thin-lens matrix ignores. Clipping produces diffraction rings: see aperture propagation for a numerical check.
  • Multimode beams: M² only scales the second-moment width. The real profile may be flat-topped or ringed, with no single wavefront curvature or Gouy phase.
  • Other effects: astigmatic or elliptical beams need separate x and y parameters. High powers cause thermal lensing and Kerr self-focusing. Absorbing media do not conserve power.

References

  • H. Kogelnik and T. Li, "Laser beams and resonators", Appl. Opt. 5, 1550–1567 (1966).
  • A. E. Siegman, Lasers, University Science Books (1986), ch. 16–17 and 20–21.
  • S. A. Self, "Focusing of spherical Gaussian beams", Appl. Opt. 22, 658–661 (1983).
  • B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019), ch. 3.
  • ISO 11146-1:2021, Lasers and laser-related equipment: test methods for laser beam widths, divergence angles and beam propagation ratios.
  • B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems II", Proc. R. Soc. A 253, 358 (1959) (non-paraxial focusing).