Gaussian Beam

Definition

A Gaussian beam is the shape of light a well-behaved laser emits. Slice across the beam and the brightness from the center outward isn't a flat-topped disk — it falls off as a bell curve, a Gaussian:

I(r, z) = I0 · (w0/w(z))² · exp(−2r² / w(z)²)

Here r is the distance from the beam axis and w(z) is the beam radius at axial position z — defined as the radius where the intensity has dropped to 1/e² ≈ 13.5% of the on-axis peak. That 1/e² edge is the "size" of the beam everyone quotes.

Unlike a flashlight, a Gaussian beam is the lowest-order solution of the paraxial wave equation. That makes it self-similar: focus it, expand it, bounce it off mirrors — it stays Gaussian. Three numbers pin it down completely.

The three numbers that define a beam

1. The waist w0. The narrowest radius the beam ever has — its pinch point. Everything else derives from w0 and the wavelength λ.

2. The Rayleigh range z_R. How far you can travel from the waist before the beam noticeably spreads:

z_R = π · w0² / λ

At z = z_R the radius has grown to √2·w0, so the spot area has exactly doubled. The beam is "in focus" over a span of about ±z_R; the full depth of focus (confocal parameter) is b = 2·z_R.

3. The divergence angle θ. Far from the waist the beam stops being flat and opens into a cone with half-angle:

θ = λ / (π · w0) = w0 / z_R

Multiply waist by divergence and the w0 cancels: w0 · θ = λ/π. This beam-parameter product is the same no matter how you reshape the beam — it's a conservation law set by the wavelength alone. You can trade a tight waist for low divergence, but never beat the product.

How the envelope grows

Put the three numbers together and you get the master equation for how wide the beam is everywhere:

w(z) = w0 · √(1 + (z / z_R)²)

This is a hyperbola. Three regimes fall out of it:

• Near field (z ≪ z_R): the square-root is ≈ 1, so w(z) ≈ w0. The beam looks collimated — a near-cylinder.
• At z = z_R: w = √2·w0. The corner of the hyperbola; beyond here spreading dominates.
• Far field (z ≫ z_R): w(z) ≈ (w0/z_R)·z = θ·z. A straight cone, as if the light radiated from a point at the waist.

The wavefronts tell a parallel story. At the waist they are flat (radius of curvature R = ∞). They curve most sharply at exactly z = z_R, where R bottoms out at 2·z_R, then flatten back toward spherical in the far field. Threaded through all of this is the Gouy phase — an extra π of phase the beam slips through as it crosses focus, which matters for mode-locked lasers and nonlinear optics.

Worked example — focusing a Nd:YAG beam

Take an Nd:YAG laser, λ = 1064 nm, and a collimated beam of radius w = 1 mm hitting a lens of focal length f = 100 mm. What spot does it make, and how deep is the focus?

Focused waist: for an ideal beam, the lens produces

w0 ≈ λ·f / (π·w)
   = (1064e-9 m × 0.100 m) / (π × 1.0e-3 m)
   ≈ 33.9 µm

Rayleigh range at the new waist:

z_R = π · w0² / λ
    = π × (33.9e-6 m)² / 1064e-9 m
    ≈ 3.39 mm

So the depth of focus b = 2·z_R ≈ 6.8 mm — that's the window in which the spot stays tight. Divergence after the waist:

θ = λ / (π·w0) = 1064e-9 / (π × 33.9e-6) ≈ 10.0 mrad ≈ 0.57°

Now feed in a real beam with M² = 1.3. The waist grows to 1.3 × 33.9 ≈ 44 µm, the depth of focus shrinks, and the divergence rises to θ = M²·λ/(π·w0). Same lens, worse beam — measurably bigger spot and shorter working range. That single factor is why cutting and welding lasers chase low M².

Real beams and the M² quality factor

No real laser is a perfect Gaussian. The M² factor (also written M-squared or "times-diffraction-limited number") quantifies the gap:

θ_real = M² · λ / (π · w0)          (divergence is M² times worse)
w0 · θ_real = M² · λ / π            (beam-parameter product scaled by M²)

An ideal TEM00 beam has M² = 1, the diffraction limit. A real beam with M² = 4 fans out four times faster and cannot be focused to a spot smaller than four times the ideal — it carries 4× the étendue. Because the beam-parameter product scales with M², it acts like an effective wavelength of M²·λ everywhere in the propagation formulas.

Source	Typical M²	Consequence
HeNe (632.8 nm)	1.0 – 1.05	Textbook Gaussian; pencil-thin over meters
Single-mode fiber laser	1.0 – 1.1	Near-ideal; focuses to a few µm
DPSS green (532 nm)	1.1 – 1.3	Excellent for fine cutting and microscopy
Edge-emitting diode (single mode)	1.1 – 1.5	Astigmatic; needs anamorphic correction
Diode bar (multimode)	20 – 100+	High power, poor focusability; pump-only use
Excimer (UV, multimode)	5 – 30	Big energy, flat-top shaping for lithography

Sizes and Rayleigh ranges in practice

How the Rayleigh range scales is the part that surprises people: it goes as the square of the waist. Halve the spot and you quarter the depth of focus. Here are representative numbers at common wavelengths.

Waist w0	Wavelength λ	Rayleigh range z_R	Divergence θ
5 mm (expanded collimated)	1064 nm	≈ 73.8 m	≈ 0.068 mrad
1 mm	1064 nm	≈ 2.95 m	≈ 0.34 mrad
0.5 mm	633 nm (HeNe)	≈ 1.24 m	≈ 0.40 mrad
50 µm	532 nm	≈ 14.8 mm	≈ 3.4 mrad
10 µm (tight focus)	532 nm	≈ 591 µm	≈ 17 mrad
1 µm (high-NA microscope)	405 nm	≈ 7.8 µm	≈ 0.13 rad

Propagating a beam — the ABCD / q-parameter method

You don't re-derive w(z) at every lens. Instead you pack waist and curvature into one complex number, the complex beam parameter q:

1/q(z) = 1/R(z) − i·λ / (π·w(z)²)
at the waist:  q0 = i·z_R

Any optical element (free space, lens, mirror) has a 2×2 ABCD ray matrix, and q transforms by the same fractional-linear rule as a ray:

q_out = (A·q_in + B) / (C·q_in + D)

Free space of length d is [[1, d], [0, 1]]; a thin lens of focal length f is [[1, 0], [−1/f, 1]]. Cascade the matrices, push q0 through, and read off the new waist and position from the imaginary and real parts. This is exactly how laser cavity designers find a stable resonator's mode in a few lines of code.

// Propagate a Gaussian beam's complex q-parameter through a thin lens.
// q is a complex number {re, im}; 1/q = 1/R - i*lambda/(pi*w^2).
const lambda = 1064e-9;            // m

function rayleigh(w0) { return Math.PI * w0 * w0 / lambda; }

// Start at a waist w0 -> q0 = i * z_R (R = infinity)
function waistToQ(w0) { return { re: 0, im: rayleigh(w0) }; }

// ABCD transform: q' = (A q + B) / (C q + D)
function abcd(q, A, B, C, D) {
  const numRe = A * q.re + B, numIm = A * q.im;
  const denRe = C * q.re + D, denIm = C * q.im;
  const den = denRe * denRe + denIm * denIm;
  return {
    re: (numRe * denRe + numIm * denIm) / den,
    im: (numIm * denRe - numRe * denIm) / den,
  };
}

const freeSpace = (q, d) => abcd(q, 1, d, 0, 1);
const thinLens  = (q, f) => abcd(q, 1, 0, -1 / f, 1);

// Read waist radius from q anywhere: w = sqrt(-lambda / (pi * Im(1/q)))
function beamRadius(q) {
  const mag2 = q.re * q.re + q.im * q.im;
  const inv1qIm = -q.im / mag2;              // Im(1/q)
  return Math.sqrt(-lambda / (Math.PI * inv1qIm));
}

let q = waistToQ(1e-3);     // 1 mm collimated waist at the lens
q = thinLens(q, 0.100);     // f = 100 mm lens
// New waist is where Re(1/q) = 0, i.e. propagate to the focus:
q = freeSpace(q, 0.0998);   // ~ just before focus
console.log((beamRadius(q) * 1e6).toFixed(1) + " um");  // ~34 um focused spot

Where Gaussian beams matter

• Laser cutting and welding. The depth of focus b = 2·z_R sets how thick a plate you can cut with one focus setting. Low M² fiber lasers win because they hold a tight spot through thick metal.
• Confocal and two-photon microscopy. The diffraction-limited waist is the resolution; the Rayleigh range is the axial sectioning thickness. Tighter focus, thinner slices.
• Optical tweezers. The intensity gradient of a tightly focused Gaussian traps micron-scale beads and cells at the waist.
• Free-space laser communication and LIDAR. Low divergence keeps power on a distant detector; θ = λ/(π·w0) tells you how big to expand the beam first.
• Laser cavity design. The stable resonator's mode is a Gaussian; the ABCD/q method finds its waist and confirms stability.
• Gravitational-wave interferometers. LIGO's kilometers-long arms run TEM00 Gaussian beams; mode-matching the beams to the cavities is a precision art.

Common mistakes and misconceptions

• Thinking a laser is a perfect parallel pencil. Every beam diverges. A "collimated" beam is just one with a large enough waist that z_R dwarfs your working distance.
• Confusing diameter with radius. w0 is a radius. The 1/e² beam diameter is 2·w0. Mixing them throws a factor of two into z_R and θ.
• Assuming a tighter focus is always better. Smaller w0 means smaller z_R (∝ w0²) and larger θ — you lose depth of focus and the beam fans out fast. There's always a trade.
• Ignoring M². Plugging the ideal λ into focusing formulas overpredicts how small a real, high-M² beam can focus. Use the effective wavelength M²·λ.
• Forgetting the 1/e² convention. Some instruments report FWHM (full width at half maximum). FWHM = 1.18·w0 in radius terms; quoting the wrong one misstates the beam size by ~18%.
• Treating the waist as where the lens sits. For a focusing lens the waist is near, but not exactly at, the focal plane — and it shifts with M² and input curvature.

Why the envelope is a hyperbola

The Gaussian beam is the fundamental solution of the paraxial Helmholtz equation. Its complex amplitude carries an explicit z-dependence in both the spot size w(z) and the wavefront radius R(z). Working through the algebra, the spot size obeys:

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

Take the square root and you have the hyperbola w(z) = w0·√(1 + (z/z_R)²). Differentiate at large z and the slope tends to w0/z_R = θ — the asymptotic cone. The radius of curvature R(z) = z·(1 + (z_R/z)²) goes to infinity both at the waist (flat) and far away (a small portion of a huge sphere), bottoming at z = z_R. The whole structure is governed by the single dimensionless ratio z/z_R, which is why scaling any beam by its own Rayleigh range collapses every Gaussian onto the same universal curve.