Near field and far field

The near field of a source is the intensity pattern at, or very close to, its emitting surface: the mode on the facet of a laser diode, the light on the cleaved end of a fiber, the illuminated area of an aperture. The far field is the pattern at a distance large enough that its shape no longer changes as it propagates; it then depends only on angle and grows in proportion to distance. In diffraction theory these are the Fresnel and Fraunhofer zones, and the far field is the Fourier transform of the near-field amplitude. A common boundary is the Fraunhofer distance $2D^2/\lambda$ for a source of diameter $D$: about 2.0 km for a 25 mm aperture at 633 nm, and only 129 µm for a 10 µm fiber mode at 1550 nm.

The boundary

At distance $L$ from an aperture of diameter $D$, the path from the edge and from the center to an on-axis point differ by about $D^2/(8L)$. When this is small compared with $\lambda$, the quadratic phase across the aperture can be dropped and the Fraunhofer approximation applies. Setting the path difference to $\lambda/16$ gives

$$L_F = \frac{2D^2}{\lambda},$$

the antenna-engineering criterion, equivalent to a Fresnel number $a^2/(\lambda L)$ of 1/8 with $a = D/2$. That entry covers the intermediate Fresnel region and its on-axis oscillations.

For a Gaussian beam the natural boundary is the Rayleigh range $z_R = \pi w_0^2/\lambda$. Within $z_R$ of the waist the beam is close to collimated and its width changes little; beyond a few $z_R$ the width grows linearly at the divergence half-angle $\theta_0 = \lambda/(\pi w_0)$. For a fiber mode with $w_0$ = 5 µm at 1550 nm in air, $z_R$ = 50.7 µm and $\theta_0$ = 0.099 rad (5.7°), so 10 mm from the fiber end the $1/e^2$ radius is already 0.99 mm. Because $2D^2/\lambda$ uses the full diameter, it is about 2.5 times $z_R$ for the same width.

A lens brings the far field to its back focal plane: in the focal plane of a lens of focal length $f$, light leaving the source at angle $\theta$ arrives at height $f\sin\theta$ for a lens designed to that mapping, so a 10 mm lens maps 30° to 5.0 mm. This is the basis of Fourier optics.

Near field and far field of laser diodes and fibers

For an edge-emitting laser, the near field is roughly 1 µm high and a few micrometers wide; the far field is the inverse, wide in the fast axis and narrow in the slow axis, so the elliptical spot appears to rotate by 90° between facet and screen, as described under fast and slow axis. With $w_0$ = 0.5 µm at 980 nm the fast-axis Rayleigh range is 0.80 µm, and with $w_0$ = 2 µm the slow-axis value is 12.8 µm, so a screen a millimeter away already records the far field. The angular widths are the far-field divergence values quoted on datasheets.

The near field shows the lateral mode, filamentation in broad-area lasers and the light distribution on a fiber end; the far field shows the divergence, the transverse-mode content, beam steering with current and the angular spread that sets coupling efficiency. For single-mode fiber, the reference method for mode field diameter is a far-field scan: the Petermann II diameter is computed from the rms width of the far-field pattern, because the far-field angles are large enough to measure accurately, while the near field, about 10 µm across, is hard to image with sub-micrometer accuracy.

Measurement

Near fields are imaged onto a camera or beam profiler through a calibrated microscope objective whose numerical aperture exceeds the source's divergence, so the full angular content is collected; the resolution, about $\lambda/(2\,\text{NA})$, approaches the size of a laser-diode mode. Far fields are measured with a rotating detector on a goniometer, with a camera at a distance well beyond $2D^2/\lambda$, or with a camera in the focal plane of a Fourier lens, as in How to Measure Laser Beam Divergence. "Near field" also names the evanescent field within a fraction of a wavelength of a surface, probed by near-field scanning optical microscopy, a separate usage.

Pitfalls

A camera placed "far away" may still be in the Fresnel zone for a wide, collimated beam: the 25 mm aperture above needs about 2 km. Truncation at an aperture adds diffraction rings to the far field, and a near-field image taken with too low an NA makes multimode structure look smoother than it is. FWHM and $1/e^2$ widths differ by a factor of 1.7 for a Gaussian, and reports should say which is used.

Common questions

What is the Fraunhofer distance?

The distance $2D^2/\lambda$ beyond which the pattern from an aperture of diameter $D$ is, to good approximation, its far-field (Fraunhofer) pattern. It scales with the square of the size.

Why is the far field the Fourier transform of the near field?

Far from the source, light reaching direction $\theta$ from different points of the aperture has a phase that varies linearly across it, at spatial frequency $\sin\theta/\lambda$; summing it is a Fourier transform.

References: J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); A. E. Siegman, Lasers (University Science Books, 1986); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).