Glossary entry

Rayleigh range (z_R)

The propagation distance from the waist of a Gaussian beam over which the beam radius increases by a factor of √2. Sets the depth of focus for any optical system.

The Rayleigh range of a Gaussian beam is

z_R \;=\; \frac{\pi w_0^2}{\lambda_0},

where $w_0$ is the beam waist radius and $\lambda_0$ is the free-space wavelength. At $z = z_R$ from the waist, the beam has expanded to $w_0 \sqrt{2}$ (intensity reduced to half on-axis).

The full depth of focus — typically defined as $2 z_R$ — is the longitudinal range over which the beam remains "nearly collimated" near the waist. Beyond $\pm z_R$ , the beam diverges at the asymptotic angle $\theta \approx \lambda_0 / (\pi w_0)$ .

Rayleigh range scales as $w_0^2$ , so small focused spots have very short depth of focus:

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 (SMF-28 MFD/2)	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

Tradeoffs. The Rayleigh range trades against minimum spot size: focusing tighter (smaller $w_0$ ) gives shorter depth of focus. For a system focusing into a sample:

Microscopy / lithography: tight focus and short Rayleigh range are desirable; alignment tolerances are matched by sample positioning resolution
Long-distance laser communication: large $w_0$ and very long $z_R$ are required; collimated beam over kilometers
Fiber coupling: $z_R$ comparable to the working distance of the coupling optic; alignment tolerance scales with $z_R$

Coupling alignment tolerance. Longitudinal alignment tolerance for fiber-to-fiber or fiber-to-PIC coupling at $1$ dB excess loss is approximately $\pm z_R$ for two waists of equal size, weakening to $\pm \sqrt{z_{R,1} z_{R,2}}$ for mismatched waists. This is the reason that high-NA lensed fiber to PIC inverse-taper coupling has $\sim 10$ μm longitudinal tolerance, while large-mode-area free-space alignment has mm-scale longitudinal tolerance.

Confocal parameter. The total range $b = 2 z_R$ is also called the confocal parameter; this is a useful figure of merit for systems where the beam needs to remain approximately collimated.

For multi-element optics: Gaussian beam parameters propagate through paraxial optical systems via ABCD matrices using the complex beam parameter

\frac{1}{q(z)} \;=\; \frac{1}{R(z)} - \frac{i \lambda_0}{\pi w^2(z)},

which transforms under an ABCD matrix as $q_2 = (A q_1 + B) / (C q_1 + D)$ . The Rayleigh range at any output plane can be read off from the imaginary part of $1/q$ .