Glossary entry

Rayleigh range

The distance from the waist of a Gaussian beam at which its radius has grown by a factor of √2, z_R = πw₀²/λ: 55 µm for the mode of standard single-mode fiber at 1550 nm and 2.0 m for a beam with a 1 mm waist radius. Twice it, the confocal parameter, is the usual depth of focus of a laser beam.

Optics & beams Optics fundamentals Lab practiceUpdated October 2026

The Rayleigh range, or Rayleigh length, of a Gaussian beam is the distance from its waist at which the beam radius has grown by a factor of $\sqrt{2}$ , so that the beam's area has doubled and its intensity on the axis has halved:

z_{R} = \frac{π w_{0}^{2} n}{M^{2} λ_{0}},

where $w_{0}$ is the waist radius, $λ_{0}$ the vacuum wavelength, $n$ the refractive index of the medium and $M^{2}$ the beam quality factor, 1 for an ideal Gaussian beam. It divides the beam into a near field within about $\pm z_{R}$ of the waist, where the beam is roughly collimated, and a far field beyond several $z_{R}$ , where the radius grows linearly at the half-angle $θ = M^{2} λ_{0} / (π n w_{0})$ ; the two are linked by $z_{R} = w_{0} / θ$ . The mode leaving standard single-mode fiber at 1550 nm, with a waist radius of 5.2 µm, has a Rayleigh range of 54.8 µm, and a beam with a 1 mm waist radius at the same wavelength one of 2.0 m.

Because $z_{R}$ grows as the square of the waist, small spots have very short Rayleigh ranges. Computed for $M^{2} = 1$ in air:

Wavelength	$w_{0}$	$z_{R}$
1550 nm	1.0 µm	2.0 µm
1550 nm	3.0 µm	18 µm
1550 nm	5.2 µm (mode of SMF-28)	55 µm
1550 nm	50 µm	5.1 mm
1550 nm	1.0 mm	2.0 m
532 nm	1.0 µm	5.9 µm
532 nm	25 µm	3.7 mm
633 nm (HeNe)	0.5 mm	1.24 m

Depth of focus

The full range $2 z_{R}$ , also called the confocal parameter, is the usual measure of the depth of focus of a laser beam. For a collimated beam focused at a working f-number $N$ , the focal length divided by the $1 / e^{2}$ beam diameter at the lens, the waist radius is $2 λ N / π$ (beam waist), and for $M^{2} = 1$

2 z_{R} = \frac{8 λ N^{2}}{π} .

f-number $N$	532 nm	1064 nm	1550 nm
5	33.9 µm	67.7 µm	98.7 µm
10	136 µm	271 µm	395 µm
20	0.54 mm	1.08 mm	1.58 mm
50	3.39 mm	6.77 mm	9.87 mm

Halving the spot size cuts the depth of focus by four. Laser machining and other work that needs the spot to stay close to its smallest size uses a fraction of this range: the radius has grown by 5% at $0.32 z_{R}$ from the waist. A beam with $M^{2} > 1$ focused to the same spot has a depth of focus $M^{2}$ times shorter.

The trade between spot size and depth of focus in laser focusing, with $M^{2}$ included, is worked through in Focusing a Laser Beam.

Curvature and phase

The radius of curvature of the wavefronts is $R (z) = z [1 + (z_{R} / z)^{2}]$ : infinite at the waist, smallest at $z = z_{R}$ , where it equals $2 z_{R}$ , and close to $z$ in the far field, where the beam looks like a spherical wave from the waist. The phase of the beam also slips against that of a plane wave by the Gouy phase $\arctan (z / z_{R})$ , $π / 4$ at the Rayleigh range and $π / 2$ from the waist to the far field. In a resonator this phase sets the frequency spacing of the transverse modes.

How far a beam stays collimated

A collimated beam stays within $\sqrt{2}$ of its waist radius for $z_{R}$ on either side of the waist, 1.24 m for a HeNe laser beam with a 0.5 mm waist radius. For a beam launched from its waist toward a target at a distance $L$ , the radius at the target, $w_{0} \sqrt{1 + (L / z_{R})^{2}}$ , is smallest when $z_{R} = L$ , which takes a waist radius of $\sqrt{λ L / π}$ and gives $\sqrt{2}$ times that at the target: at 1550 nm, 22 mm for 1 km and 70 mm for 10 km. Longer free-space links therefore need larger apertures, since the beam cannot stay collimated over more than its Rayleigh range.

Alignment tolerance

The Rayleigh range also sets how closely two beams have to be positioned along the axis to couple. Two Gaussian modes with waist radii $w_{1}$ and $w_{2}$ , whose waists are a distance $d$ apart along the axis, couple with efficiency

η = \frac{4}{(w_{1} / w_{2} + w_{2} / w_{1})^{2} + {(λ d / π w_{1} w_{2})}^{2}},

as Joyce and DeLoach set out. For equal waists a separation of $1.02 z_{R}$ costs 1 dB and $0.70 z_{R}$ costs 0.5 dB. For unequal waists, 1 dB beyond the loss from the size mismatch is reached at $0.51 (z_{R 1} + z_{R 2})$ , so the larger of the two Rayleigh ranges sets the tolerance. Two standard single-mode fibers at 1550 nm can be 56 µm apart for 1 dB, while a lensed fiber with a 2.5 µm spot, coupled to a silicon inverse taper with a 2.6 µm mode, tolerates 3.4 µm; its lateral tolerance, 0.6 µm for 1 dB, is tighter still (mode mismatch loss).

Through optical systems

Through lenses, mirrors and free space the beam is followed with the complex beam parameter $q$ , defined by

\frac{1}{q} = \frac{1}{R} - \frac{i λ}{π w^{2}},

which at a distance $z$ from a waist is $q = z + i z_{R}$ . A system with ABCD matrix elements $A$ , $B$ , $C$ , $D$ transforms it as $q_{2} = (A q_{1} + B) / (C q_{1} + D)$ , and the imaginary part of $q_{2}$ is the Rayleigh range of the beam that leaves the system, its real part the distance past that beam's waist, negative while the waist still lies ahead. The Gaussian Beam Calculator gives $z_{R}$ and the depth of focus $2 z_{R}$ for any waist, wavelength and $M^{2}$ , plots $w (z)$ with the Rayleigh range marked, and follows the beam through a lens.

Common questions

What is the Rayleigh range?

The distance from the waist of a Gaussian beam at which the beam radius has grown by $\sqrt{2}$ , $z_{R} = π w_{0}^{2} / λ$ . Within about $\pm z_{R}$ of the waist the beam is roughly collimated; far beyond it the beam spreads at a constant angle.

Is the Rayleigh length the same as the Rayleigh range?

Yes, the two names are used for the same distance. The confocal parameter is twice it.

For a laser beam the depth of focus is usually taken as $2 z_{R}$ , the range over which the radius stays within $\sqrt{2}$ of the waist: 67.7 µm for a 1064 nm beam focused at f/5. A tighter tolerance on the spot size gives a shorter usable depth.

How does M² affect the Rayleigh range?

For a given waist radius the Rayleigh range is divided by $M^{2}$ , since the beam spreads $M^{2}$ times faster. For a given focusing f-number the spot is $M^{2}$ times larger and the depth of focus $M^{2}$ times longer.

References: H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966). A. E. Siegman, Lasers (University Science Books, 1986). W. B. Joyce and B. C. DeLoach, "Alignment of Gaussian beams," Applied Optics 23, 4187 (1984). ISO 11146-1:2021, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles and beam propagation ratios.