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Knowledge Hubwireless communication basics8. Reactive Near Field, Radiative Near Field, and Far Field

wireless communication basics learning note

8. Reactive Near Field, Radiative Near Field, and Far Field

Identify electromagnetic field regions, calculate commonly used boundaries, and understand how distance changes coupling, wavefront shape, beam response, and channel modelling.

Why the Field Region Changes the Model

An antenna does not produce the same spatial behaviour at every distance. Very close to the antenna, stored electric and magnetic energy dominates. Farther away, energy radiates but the wavefront can remain strongly curved. At a sufficiently large distance, the field approaches a locally plane wave and its angular pattern becomes almost independent of range.

Original diagram of reactive near-field, radiative near-field, and far-field regions
Original diagram: distance from an aperture changes the dominant physics from reactive coupling, through Fresnel range-angle behaviour, to the approximately plane-wave Fraunhofer region.

These regions are gradual modelling regimes rather than physical walls. The usual boundaries are engineering approximations based on allowable phase error and antenna size.

References for this section3

The Three Common Regions

Let DD be the largest physical dimension of the antenna or aperture and λ\lambda the wavelength. For an electrically large aperture, commonly used boundaries are

rreactive≈0.62D3λr_{\mathrm{reactive}} \approx 0.62\sqrt{\frac{D^3}{\lambda}}

and

rFF≈2D2λr_{\mathrm{FF}} \approx \frac{2D^2}{\lambda}

This produces the approximate classification

{r<0.62D3/λ,reactive near field,0.62D3/λ≲r<2D2/λ,radiative near field,r≳2D2/λ,far field.\begin{cases} r < 0.62\sqrt{D^3/\lambda}, & \text{reactive near field},\\[4pt] 0.62\sqrt{D^3/\lambda}\lesssim r < 2D^2/\lambda, & \text{radiative near field},\\[4pt] r \gtrsim 2D^2/\lambda, & \text{far field}. \end{cases}

For electrically small antennas, other conventions such as a boundary near λ/(2π)\lambda/(2\pi) are often more meaningful. The formulas should therefore be applied with the antenna type, aperture, and required accuracy stated explicitly.

References for this section3

Reactive Near Field

The reactive near field lies closest to the antenna. Energy is repeatedly stored and returned by the local electric and magnetic fields instead of flowing steadily outward as radiation. For an elementary radiator, field expressions contain terms that decay approximately as

1r3,1r2,and1r\frac{1}{r^3},\qquad \frac{1}{r^2},\qquad \text{and}\qquad \frac{1}{r}

The 1/r31/r^3 and 1/r21/r^2 components are most influential at short range; the radiating 1/r1/r component dominates farther away. Electric and magnetic fields need not have the free-space plane-wave ratio E/H=η0E/H=\eta_0 in this region.

Expected response:

  • strong inductive or capacitive coupling;
  • large sensitivity to nearby objects and probe placement;
  • antenna impedance and resonance can change when an object enters the field;
  • power does not follow a simple far-field 1/r21/r^2 density law;
  • angle-only antenna patterns are not a complete description.

Near-field communication, wireless charging, RFID coupling, antenna detuning, and exposure measurements can all involve this regime.

References for this section3

Radiative Near Field or Fresnel Region

In the radiative near field, propagating energy dominates, but the wavefront curvature across the aperture is still significant. A point at range rr and angle cannot be represented accurately by angle alone.

For an array element at position pn\mathbf{p}_n and a user at p\mathbf{p}, the exact propagation distance is

rn=∥p−pn∥r_n=\left\|\mathbf{p}-\mathbf{p}_n\right\|

and a spherical-wave channel term can be modelled as

hn∝1rne−j2πλrnh_n \propto \frac{1}{r_n} e^{-j\frac{2\pi}{\lambda}r_n}

Both amplitude and phase therefore vary across a sufficiently large aperture.

Expected response:

  • beams can focus at a chosen range and angle;
  • the apparent beam pattern changes with observation distance;
  • spherical-wave steering provides range information for localization;
  • far-field plane-wave codebooks can suffer focusing loss;
  • different parts of a very large array may observe meaningfully different path lengths or visibility.

This region is increasingly important for massive arrays, extremely large aperture arrays, mmWave systems, and large reconfigurable intelligent surfaces.

References for this section3

How the Near Field Reveals Range

Start with a straight array along the xx-axis. Its centre is the origin O=(0,0)O=(0,0). The nnth element is at

pn=(xn,0)\mathbf{p}_n=(x_n,0)

and the user is at point PP. We measure angle θ\theta from the array's broadside direction, the positive zz-axis. The user's Cartesian coordinates are therefore

P=(rsin⁡θ, rcos⁡θ)P=(r\sin\theta,\ r\cos\theta)

where rr is the distance from the array centre to the user.

Original labelled geometry showing array centre O, element coordinate x n, user distance r, element-to-user distance r n, angle theta, and the user's horizontal and vertical coordinates
Original coordinate geometry: $r$ is measured from the array centre, while $r_n$ is measured from element $n$. The angle $\theta$ is measured from broadside, so the user's horizontal and vertical coordinates are $r\sin\theta$ and $r\cos\theta$.

The three lengths xnx_n, rr, and rnr_n form a triangle. The horizontal separation between element nn and the user is rsin⁡θ−xnr\sin\theta-x_n, while the vertical separation is rcos⁡θr\cos\theta. Pythagoras gives

rn2=(rsin⁡θ−xn)2+(rcos⁡θ)2r_n^2 = (r\sin\theta-x_n)^2+(r\cos\theta)^2

Expanding the squares,

rn2=r2sin⁡2θ−2rxnsin⁡θ+xn2+r2cos⁡2θ=r2−2rxnsin⁡θ+xn2\begin{aligned} r_n^2 &=r^2\sin^2\theta-2rx_n\sin\theta+x_n^2+r^2\cos^2\theta\\ &=r^2-2rx_n\sin\theta+x_n^2 \end{aligned}

because sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1. Taking the positive square root gives the exact distance

rn(r,θ)=r2+xn2−2rxnsin⁡θr_n(r,\theta) = \sqrt{r^2+x_n^2-2rx_n\sin\theta}

This equation already contains the central idea: changing either rr or θ\theta changes the set of element distances {rn}\{r_n\}.

Using the array centre as a phase reference, a narrowband spherical-wave steering-vector entry can be written as

an(r,θ)=rrnexp⁡ ⁣[−jk(rn−r)],k=2πλa_n(r,\theta) = \frac{r}{r_n} \exp\!\left[-jk\bigl(r_n-r\bigr)\right], \qquad k=\frac{2\pi}{\lambda}

The factor r/rnr/r_n represents relative amplitude variation, while the exponential describes relative phase across the aperture. When rr is not very large compared with the aperture, both depend on range and angle.

Original diagram showing two users at the same angle but different ranges producing different spherical phase curvature across an array
Original diagram: two users can share the same direction yet produce different path-length curvature across a large aperture. The near-field array records a distinct spatial phase fingerprint for each range.
References for this section3

The Fresnel Expansion Separates Angle and Range

The approximation

rn≈r−xnsin⁡θ+xn2cos⁡2θ2rr_n \approx r -x_n\sin\theta +\frac{x_n^2\cos^2\theta}{2r}

comes from a second-order Taylor expansion; it is not an assumed formula. Begin by taking r2r^2 outside the square root:

rn=r1−2xnrsin⁡θ+xn2r2=r1+u\begin{aligned} r_n &= r\sqrt{1-2\frac{x_n}{r}\sin\theta+\frac{x_n^2}{r^2}}\\ &=r\sqrt{1+u} \end{aligned}

with

u=−2xnrsin⁡θ+xn2r2u=-2\frac{x_n}{r}\sin\theta+\frac{x_n^2}{r^2}

When ∣xn∣/r|x_n|/r is sufficiently small, use

1+u≈1+u2−u28\sqrt{1+u} \approx 1+\frac{u}{2}-\frac{u^2}{8}

and retain terms only up to order (xn/r)2(x_n/r)^2. The first correction is

u2=−xnrsin⁡θ+xn22r2\frac{u}{2} = -\frac{x_n}{r}\sin\theta +\frac{x_n^2}{2r^2}

For u2u^2, the only required second-order term comes from squaring the linear part of uu:

−u28≈−xn22r2sin⁡2θ-\frac{u^2}{8} \approx -\frac{x_n^2}{2r^2}\sin^2\theta

Combining the two corrections and multiplying by rr,

rn≈r[1−xnrsin⁡θ+xn22r2−xn22r2sin⁡2θ]=r−xnsin⁡θ+xn22r(1−sin⁡2θ)=r−xnsin⁡θ+xn2cos⁡2θ2r\begin{aligned} r_n &\approx r\left[ 1-\frac{x_n}{r}\sin\theta +\frac{x_n^2}{2r^2} -\frac{x_n^2}{2r^2}\sin^2\theta \right]\\ &= r-x_n\sin\theta +\frac{x_n^2}{2r}\left(1-\sin^2\theta\right)\\ &= r-x_n\sin\theta +\frac{x_n^2\cos^2\theta}{2r} \end{aligned}

The denominator is 2r2r, and the numerator is xn2cos⁡2θx_n^2\cos^2\theta. Dimensionally, xn2/(2r)x_n^2/(2r) has units of length, as every term in a distance approximation must.

The three terms now have clear meanings:

r⏟common distance−xnsin⁡θ⏟linear variation: anglexn2cos⁡2θ2r⏟quadratic curvature: range\underbrace{r}_{\text{common distance}} \qquad \underbrace{-x_n\sin\theta}_{\text{linear variation: angle}} \qquad \underbrace{\frac{x_n^2\cos^2\theta}{2r}}_{\text{quadratic curvature: range}}

The linear term creates the familiar progressive phase used for far-field direction finding. The quadratic term measures wavefront curvature and contains 1/r1/r. Two users at the same angle but different ranges therefore generate different quadratic phase profiles.

For equally spaced elements with spacing dd, this curvature can be seen through the second distance difference:

rn+1−2rn+rn−1≈d2cos⁡2θrr_{n+1}-2r_n+r_{n-1} \approx \frac{d^2\cos^2\theta}{r}

The corresponding second phase difference is approximately

Δ2ϕn≈−kd2cos⁡2θr\Delta^2\phi_n \approx -k\frac{d^2\cos^2\theta}{r}

The linear angle term disappears from this second difference, while the 1/r1/r curvature term remains. A nearer user produces stronger bending across the phase samples; a farther user produces a flatter profile. This is the spatial cue that lets a calibrated near-field array distinguish range.

As rr becomes large, the quadratic term becomes too small to resolve. The steering vector then approaches

an(r,θ)≈ejkxnsin⁡θa_n(r,\theta) \approx e^{jkx_n\sin\theta}

apart from a common phase and amplitude. The normalized far-field spatial signature contains angle but essentially no range information. Range may still be measured through propagation delay when adequate signal bandwidth and timing synchronization are available; it is the array curvature cue that disappears.

References for this section3

Range–Angle Focusing Response

An array can test a candidate point (r0,θ0)(r_0,\theta_0) by matching the measured channel against its spherical-wave steering vector. A normalized focusing response is

G(r0,θ0;r,θ)=∣aH(r0,θ0)a(r,θ)∣2N2G(r_0,\theta_0;r,\theta) = \frac{\left| \mathbf{a}^{\mathrm{H}}(r_0,\theta_0) \mathbf{a}(r,\theta) \right|^2}{N^2}

The response approaches one when the candidate and true range–angle pair produce aligned phases. An incorrect range leaves a residual quadratic phase error, so the element contributions do not add fully coherently.

For a fixed range hypothesis r0r_0, the residual phase at element nn is approximately

Δϕn≈−kxn2cos⁡2θ2(1r−1r0)\Delta\phi_n \approx -\frac{kx_n^2\cos^2\theta}{2} \left(\frac{1}{r}-\frac{1}{r_0}\right)

This equation shows why larger electrical apertures improve range discrimination: the mismatch grows with xn2/λx_n^2/\lambda. A useful scaling for the depth of a focused region is

Δrfocus∝λr2D2\Delta r_{\mathrm{focus}} \propto \frac{\lambda r^2}{D^2}

where the proportionality constant depends on the chosen loss or resolution criterion, array illumination, and geometry. Range separation becomes harder at longer distance and easier with a larger aperture or shorter wavelength.

References for this section3

What Is Needed in Practice

Near-field range estimation is not automatic merely because a user lies inside a nominal boundary. The receiver also needs:

  • a calibrated array with known element positions and phase offsets;
  • enough aperture for measurable curvature relative to noise and hardware error;
  • a channel model that preserves element-wise spherical distances;
  • sufficient SNR and distinguishable propagation paths;
  • control of phase ambiguity, synchronization error, and mutual coupling;
  • multiple frequencies, snapshots, or prior information when a single narrowband signature is ambiguous.

Wideband delay and near-field curvature are complementary. Delay estimates absolute path length from frequency-dependent timing, while spatial curvature estimates how distance changes across the aperture. Combining both can improve 3D localization and reduce range–angle ambiguity.

References for this section3

Far Field or Fraunhofer Region

In the far field, the wavefront curvature across the aperture is small enough that the incident field can be approximated locally as a plane wave. Expanding the element distance gives

rn≈r−u^Tpnr_n \approx r-\widehat{\mathbf{u}}^{\mathsf{T}}\mathbf{p}_n

where u^\widehat{\mathbf{u}} is a unit direction vector. The common range rr contributes nearly the same factor to every element, while relative phase depends primarily on direction.

For free-space propagation, the time-average power density approximately follows

S(r)=PtGt4πr2S(r) = \frac{P_{\mathrm{t}}G_{\mathrm{t}}}{4\pi r^2}

and the electric and magnetic fields approach

∣E∣∣H∣≈η0≈377 Ω\frac{|E|}{|H|}\approx\eta_0\approx377\ \Omega

Expected response:

  • angular steering vectors accurately describe the array response;
  • normalized radiation-pattern shape is nearly independent of distance;
  • electric and magnetic fields are transverse and approximately in phase;
  • field amplitude decays approximately as 1/r1/r and power density as 1/r21/r^2;
  • Friis transmission and conventional angle-only beamforming become appropriate under their other assumptions.
References for this section3

Worked Boundary Example at 28 GHz

Consider a square aperture with largest dimension D=0.50 mD=0.50\ \mathrm{m} at f=28 GHzf=28\ \mathrm{GHz}. Its wavelength is

λ=cf≈10.7 mm\lambda=\frac{c}{f}\approx10.7\ \mathrm{mm}

The approximate reactive boundary is

rreactive≈0.620.5030.0107≈2.12 mr_{\mathrm{reactive}} \approx 0.62\sqrt{\frac{0.50^3}{0.0107}} \approx2.12\ \mathrm{m}

and the Fraunhofer distance is

rFF≈2(0.50)20.0107≈46.7 mr_{\mathrm{FF}} \approx \frac{2(0.50)^2}{0.0107} \approx46.7\ \mathrm{m}

Using these approximations, a point at 1 m lies in the reactive near-field range, a point at 10 m lies in the radiative near field, and a point at 60 m lies in the far field. A smaller aperture at the same frequency would have much shorter boundaries because both expressions depend strongly on DD.

References for this section3

What the Receiver or User Observes

RegionDominant spatial modelWhat changes with positionSuitable response model
Reactive near fieldStrong local electric/magnetic couplingImpedance, coupling, field balance, amplitude, and phaseFull-wave or coupling model
Radiative near fieldSpherical radiating waveRange and angle both change array responseElement-wise spherical distance and focusing
Far fieldApproximately plane waveDirection dominates normalized array responseAngular steering vector and Friis-type link model

A user does not cross a sharp boundary and suddenly see a different signal. Instead, model error grows gradually as an inappropriate approximation is used. The required boundary can also move depending on aperture illumination, scan angle, bandwidth, acceptable phase error, and measurement objective.

References for this section3

Wideband and Practical Considerations

The boundaries depend on wavelength, so a wideband system does not have exactly one electrical boundary across its entire band. Hardware enclosures, ground planes, mutual coupling, and nearby scatterers also alter the ideal response. For accurate antenna measurements, standards may impose a larger separation than the simple 2D2/λ2D^2/\lambda rule or use near-field scanning followed by a near-to-far-field transformation.

For communication modelling, first calculate the boundary, then test whether a plane-wave channel gives acceptable phase error across the aperture. If not, use spherical-wave propagation and preserve each element-to-user distance.

References for this section3

Takeaway

The reactive near field is dominated by stored energy and coupling. The radiative near field carries outward power but preserves wavefront curvature, enabling range-angle focusing. The far field supports an approximately plane-wave, angle-based description. Aperture size, wavelength, distance, and required accuracy decide which model produces a trustworthy response.

References for this section3

Complete references and further reading