Radiation, Antennas and Scattering

Introduction and context

Time-varying electromagnetic fields can detach from their sources and transport energy through space. This process—radiation—is central to wireless communication, radar, remote sensing, EMC, biomedical telemetry, and optical systems. Antennas provide controlled conversion between guided waves and radiated waves; scattering describes the re-radiation produced when an incident field interacts with an object.

This chapter uses sinusoidal steady state with phasors having time dependence e^jωt. With this convention, Maxwell’s equations in a homogeneous, isotropic, source-free region are

where E and H are phasor electric and magnetic fields, ω is angular frequency (rad/s), and ε and μ are permittivity and permeability. The intrinsic impedance and wavenumber are η = √(μ/ε) and k = ω√(με) = 2π/λ.

Learning objectives

After studying this chapter, the reader should be able to:

1. distinguish reactive, induction, and radiating field regions;
2. derive the far field, power density, radiation resistance, and directivity of a short dipole;
3. interpret antenna gain, efficiency, effective aperture, polarization, and impedance matching;
4. apply the Friis transmission equation with correct polarization and mismatch conventions;
5. explain aperture, array, horn, reflector, and modern antenna technologies;
6. derive and use Rayleigh scattering cross sections and the optical theorem;
7. evaluate practical antenna and scattering systems, including limitations and measurement issues.

1. Radiation from localized sources

A localized current distribution radiates because accelerated charge produces fields that decrease as 1/r, rather than only the near-field terms that decrease as 1/r² or 1/r³. For a finite source, the retarded vector potential is

For an electrically small source, a ≪ λ, observation in the far zone r ≫ 2a²/λ permits R ≈ r − r̂·r′. The phase factor then contains the source’s directional Fourier transform. This is the foundation of antenna patterns and diffraction.

The radiating field is transverse: in the far zone, E ⟂ r̂, H ⟂ r̂, and H = (1/η)(r̂ × E). The time-average Poynting vector is

and its radial component represents outward power flow. The asterisk denotes complex conjugation.

Field regions

For an antenna with largest dimension D, commonly used engineering boundaries are:

These are engineering criteria, not universal physical discontinuities. For electrically small antennas, the reactive boundary is often governed by dimensions comparable with a fraction of a wavelength.

2. Hertzian dipole: field derivation and power

Consider an infinitesimal current element of length l, carrying uniform phasor current I₀, with l ≪ λ. Its electric dipole moment amplitude is related to current by I₀l = jωp₀ under the selected phasor convention. A standard spherical-coordinate solution gives

A clearer complete form, avoiding ambiguity in the near-field magnetic component, is

and

The exact signs of reactive terms depend on the adopted source and phasor convention; the 1/r far-field terms and all power results below are convention independent. In the radiation zone, kr ≫ 1,

Thus the far field is maximum broadside to the wire (θ = π/2) and zero along its axis. The radiation intensity is

because ⟨Sr⟩ = |Eθ|²/(2η). Integrating over solid angle,

Equating this to (1/2)I₀²Rᵣ defines radiation resistance:

for an ideal Hertzian dipole in free space. The formula is applicable only when the current element is electrically short and the current is treated as uniform.

The maximum radiation intensity is Umax = η(kI₀l)²/(32π²). Since the average radiation intensity is Prad/(4π), directivity is

A short, practical center-fed dipole has a triangular rather than uniform current distribution and consequently a different radiation resistance and pattern. It must not be assigned the Hertzian result without qualification.

Worked example: short-dipole power

A Hertzian dipole in free space has l = 0.01λ and rms current I = 1 A. Then

Rᵣ = 80π²(0.01)² = 0.0790 Ω.

Because rms current is used, Prad = I²Rᵣ = 79.0 mW. If peak current had been used, the factor 1/2 would be required. This distinction is essential in antenna laboratory measurements.

3. Antenna parameters and reciprocity

An antenna is characterized by several related but nonidentical quantities.

• Radiation pattern: angular variation of field, power, or radiation intensity, normally normalized to its maximum.
• Beamwidth: half-power beamwidth is the angular separation between points 3 dB below the peak; null-to-null beamwidth uses adjacent nulls.
• Radiation intensity: U = r²⟨Sr⟩, in W/sr.
• Directivity: D(θ,φ) = 4πU(θ,φ)/Prad. It excludes ohmic and dielectric loss.
• Efficiency: ηₐ = Prad/Pin, where Pin is accepted input power. It includes conductor, dielectric, and other antenna losses.
• Gain: G = ηₐD. Gain includes efficiency but not transmitter or receiver mismatch unless explicitly stated.
• Polarization: the locus and orientation of the electric-field vector at a fixed observation point.
• Effective aperture: for a receiving antenna in a matched polarization condition, Aₑ = λ²G/(4π).

Reciprocity states that a linear, passive, time-invariant antenna in a reciprocal medium has the same transmitting and receiving pattern and polarization response, when corresponding ports and directions are interchanged. Ferrite biasing, active circuits, nonlinear materials, or time variation can violate the assumptions.

A useful receive relation is

for a plane wave of power density S and matched polarization. Here P_av is available received power, not necessarily delivered load power.

Polarization mismatch

Let pᵢ and pᵣ be unit complex polarization vectors for incident and receiving antennas. The polarization loss factor is

where pᵣ* is the complex conjugate transpose. The conjugation is essential for elliptical and circular polarization. For orthogonal circular polarizations, PLF is zero; for identical circular polarization, it is one. For two linearly polarized vectors separated by angle ψ, PLF = cos²ψ.

4. Link budget and Friis transmission

In free space, two antennas separated by distance R in the far field have

where Pt,acc is power accepted by the transmitting antenna and Pr,av is available power at the receiving antenna. The equation assumes free-space propagation, far-field separation, aligned boresight directions, no multipath, and gains defined consistently for the selected polarization.

If Pt,del is generator-delivered power and the transmitter antenna has power-transfer efficiency ηmt, then Pt,acc = ηmt Pt,del. If a load receives delivered rather than available power, let ηmr be the receiver load-transfer efficiency; then

For a receiver reflection coefficient Γr referenced to its antenna port, ηmr = 1 − |Γr|² under a conjugate-matched source normalization. These explicit factors replace any undefined generic mismatch term. Transmitter mismatch must likewise be represented by ηmt = 1 − |Γt|² only when its reference and available-power definition make that relation valid.

Corrected worked example: Friis equation

Let Pt,acc = 1 W, Gt = Gr = 1, f = 2.4 GHz, R = 100 m, and PLF = 1. The wavelength is

λ = c/f = 0.125 m. Therefore,

Pr,av/Pt,acc = (0.125/(4π×100))² = 9.895×10⁻⁹.

This numerical choice gives approximately −80.0 dB. To illustrate the reviewed reference result, use R = 251.2 m with the same frequency and gains:

Pr/Pt = (0.125/(4π×251.2))² ≈ 1.58×10⁻⁹.

Thus for 1 W, Pr ≈ 1.58 nW, and the ratio is 10log₁₀(1.58×10⁻⁹) ≈ −88.0 dB. No mismatch or polarization loss is present in this calculation.

5. Antenna types and arrays

Wire antennas include monopoles, dipoles, loops, and helical antennas. A quarter-wave monopole over a perfect conducting ground plane uses image theory: the image current creates the missing half of a dipole in the upper half-space. Its ideal radiation resistance is approximately 36.5 Ω, although finite ground planes and loading change this value.

Aperture antennas—horns, slots, and reflectors—are effective at microwave frequencies. A uniformly illuminated aperture of area A has approximate directivity

with an aperture efficiency factor ηap in practical designs: G ≈ ηap4πA/λ². Taper, spillover, phase error, blockage, and polarization loss reduce efficiency.

An array factor separates element and geometry effects:

where wₙ is the complex excitation, rₙ is element position, and βₙ is excitation phase. Uniform weighting gives narrow main beams but higher sidelobes; amplitude taper lowers sidelobes at the cost of beamwidth and directivity. Mutual coupling means that the actual element currents may differ from prescribed generator excitations, so array-factor multiplication is an approximation unless coupling is modeled.

Phased arrays electronically steer beams by controlling excitation phase. Research systems increasingly use massive MIMO, metamaterial-inspired apertures, reconfigurable intelligent surfaces, and integrated antenna-in-package structures. These technologies introduce calibration, thermal, nonlinear, and manufacturing challenges.

6. Scattering fundamentals

When an incident field encounters a dielectric, conductor, or composite object, induced charges and currents reradiate. The total field is the sum of incident and scattered fields:

The far-zone scattered field is often written

where f is the scattering amplitude, with dimensions of length. The differential scattering cross section is

and total scattering cross section is σsca = ∫|f|²dΩ. The extinction cross section includes scattering and absorption:

Cross sections have units of area and are ratios of removed or redirected power to incident intensity; they can exceed the geometric area because of resonant interaction.

Rayleigh scattering and polarizability

For a small lossless dielectric sphere of radius a, relative permittivity εr, and size parameter x = ka ≪ 1, the quasistatic electric polarizability in SI units is

for a sphere in free space. The induced dipole is p = αEinc. The Rayleigh total scattering cross section is

or, for a lossless dielectric sphere,

Thus scattering varies as a⁶ and λ⁻⁴. For sufficiently small particles, short wavelengths scatter much more strongly; this explains the blue appearance of clear daytime sky when molecular absorption is weak.

If the polarizability is complex, absorption must be included. With the stated phasor convention, the optical theorem is

where f(0) is the forward scattering amplitude. For an electric dipole, a consistent convention gives

and

with passivity requiring σabs ≥ 0. The exact sign of the imaginary part changes if the opposite time convention is used; the physical cross sections do not.

Beyond Rayleigh: Mie and geometric regimes

For x not much smaller than unity, multipoles of many orders contribute and Mie theory is required for spheres. When the object is many wavelengths across, physical optics, geometrical optics, uniform theory of diffraction, or numerical integral-equation methods may be more efficient. Conducting objects can exhibit specular reflection, edge diffraction, creeping waves, and resonances. No single Rayleigh formula is valid across these regimes.

7. Numerical and experimental methods

Method of Moments (MoM) converts integral equations into a matrix system for surface or volume currents. Finite Element Method (FEM) discretizes differential equations in bounded domains and is effective for complex materials. Finite-Difference Time-Domain (FDTD) advances fields in time on a grid and is broadband but subject to the Courant stability condition. Physical optics is efficient for electrically large smooth reflectors but neglects some edge and multiple-scattering effects.

Reliable simulation requires mesh convergence, physically appropriate boundary conditions, port calibration, and validation against analytical limiting cases. Measurements commonly use anechoic chambers, near-field scanning with near-to-far-field transformation, reverberation chambers, or compact antenna test ranges. Cable loss, chamber reflections, probe calibration, dynamic range, and positioning errors must be included in uncertainty budgets.

8. Illustrative applied case: 5.8 GHz industrial sensor link

Background. An industrial sensor transmits to a gateway 30 m away. The gateway uses a directional antenna, while the sensor uses a small linearly polarized antenna.

Problem. Determine whether a nominal link budget is plausible and identify dominant engineering risks.

Analysis. Let f = 5.8 GHz, so λ ≈ 0.0517 m. Suppose accepted transmit power is 10 mW, transmitter gain is 2 dBi, receiver gain is 12 dBi, and polarization alignment is 20°. The polarization factor is cos²20° ≈ 0.883. Free-space path factor is

(λ/(4πR))² = (0.0517/(4π×30))² ≈ 1.88×10⁻⁸.

Linear gains are Gt ≈ 1.585 and Gr ≈ 15.85. Hence

Pr,av ≈ 0.01×1.585×15.85×1.88×10⁻⁸×0.883 ≈ 4.18×10⁻⁹ W,

or approximately −53.8 dBm. If receiver mismatch delivers only 80% of available power, delivered power is approximately −54.8 dBm.

Approach and solution. The design team should verify Fresnel-zone clearance, account for factory multipath and metal reflections, measure antenna patterns in the installed enclosure, and include fading margin. A polarization rotation caused by mounting can be more consequential than a small gain discrepancy.

Outcome. The free-space estimate indicates feasibility only when receiver sensitivity is comfortably below −54.8 dBm after implementation losses. The result is not a guarantee of coverage because Friis excludes multipath, shadowing, and obstruction.

Learning points. A link budget is a model with explicit assumptions; accepted, available, and delivered powers must not be interchanged. Polarization and installation environment can dominate nominal antenna specifications.

9. Limitations, governance, and professional practice

Antenna systems may cause human-exposure concerns, interference, unlawful emissions, and privacy risks when used for sensing or surveillance. Engineers must comply with national spectrum allocations, electromagnetic-exposure limits, equipment certification, and applicable EMC standards. Test data should be traceable, uncertainty should be reported, and claims should distinguish simulation, calibrated measurement, and extrapolation.

AI-assisted electromagnetic optimization can search element geometries, surrogate models, and beamforming weights, but it does not replace Maxwell-equation verification, mesh convergence, or safety review. Training-data leakage, undocumented objective functions, and designs that exploit simulation artifacts are governance risks. Best practice is to preserve reproducible scripts, versioned material data, independent validation, and human approval for deployment.

Chapter Summary

Radiation is the outward transport of energy by time-varying fields. In the far zone, electric and magnetic fields are transverse and related by intrinsic impedance. Derivation of the Hertzian dipole fields leads to the sin²θ pattern, total radiated power, radiation resistance Rᵣ = 80π²(l/λ)² Ω, and directivity 1.5. Antenna gain combines directivity with efficiency, while effective aperture connects receiving power to incident power density. Friis transmission requires consistent definitions of accepted, available, and delivered power; polarization coupling for complex vectors uses |pᵣ*·pᵢ|². Scattering is described through amplitudes and cross sections. In the Rayleigh regime, a small dielectric sphere has σsca ∝ a⁶λ⁻⁴, while the optical theorem relates extinction to forward scattering and ensures absorption is accounted for. Practical design requires numerical validation, measurement, regulatory compliance, and explicit uncertainty analysis.

Key Terms

Radiation: Electromagnetic energy carried away from a localized source.

Hertzian dipole: Infinitesimal current element used as a fundamental radiation model.

Radiation resistance: Equivalent resistance representing radiated power.

Radiation intensity: Power radiated per unit solid angle.

Directivity: Ratio of radiation intensity in a direction to its angular average.

Antenna gain: Directivity multiplied by antenna efficiency.

Effective aperture: Receiving area-equivalent parameter related to gain.

Polarization: Temporal and spatial orientation behavior of the electric field.

Polarization loss factor: Power coupling factor between incident and receiving polarizations.

Friis equation: Far-field free-space relation between transmitted and received powers.

Mismatch loss: Power not transferred because of reflection at a port.

Reactive near field: Region dominated by stored, nonpropagating energy.

Far field: Region in which fields approximate a locally plane wave.

Array factor: Pattern contribution due to element positions and excitations.

Scattering amplitude: Far-field coefficient describing reradiated field strength.

Scattering cross section: Equivalent area quantifying redirected power.

Extinction cross section: Cross section for total removal from the incident beam.

Rayleigh scattering: Small-particle scattering regime with size parameter much less than one.

Optical theorem: Relation between extinction and forward scattering amplitude.

Mie theory: Exact spherical-particle solution involving multipole expansions.

Review Questions

Short Answer Questions

1. Why do radiating fields vary as 1/r while reactive fields decrease more rapidly?
2. State the assumptions under which the Hertzian-dipole radiation-resistance formula is valid.
3. Distinguish directivity, efficiency, and gain.
4. Why is the conjugate required in the complex polarization inner product?
5. Define scattering, absorption, and extinction cross sections.

Long Answer Questions

1. Derive the far-zone fields, radiation intensity, total radiated power, radiation resistance, and directivity of a Hertzian dipole.
2. Explain the Friis transmission equation and carefully distinguish accepted, available, and delivered power.
3. Compare wire, aperture, reflector, and phased-array antennas, including their principal limitations.
4. Derive the Rayleigh scattering cross section of a small dielectric sphere from its polarizability.
5. Explain the optical theorem and discuss the transition from Rayleigh to Mie and geometric scattering.

Analytical / Application Questions

1. A short dipole has l = 0.02λ and carries 0.5 A rms in free space. Calculate its radiation resistance and radiated power, stating all assumptions.
2. A link has 6 dBi transmit gain, 14 dBi receive gain, 2 GHz frequency, and 1 km separation. Determine the free-space path factor and explain how a 45° linear polarization misalignment changes the result.
3. A dielectric sphere has ka = 0.05. Evaluate whether Rayleigh theory is initially appropriate and discuss which additional parameters are required to calculate its cross section.

Multiple Choice Questions with Answers

Question 1

For a Hertzian dipole, the far-zone electric-field magnitude varies angularly as:

A. sinθ

B. cosθ

C. sin²θ

D. 1/sinθ

Correct Answer: A. sinθ

Question 2

The radiation resistance of a Hertzian dipole of length l in free space is:

A. 80π²(λ/l)² Ω

B. 80π²(l/λ)² Ω

C. 120π(l/λ) Ω

D. λ/(80π²l) Ω

Correct Answer: B. 80π²(l/λ)² Ω

Question 3

For complex unit polarization vectors, the polarization loss factor is:

A. |pᵣ · pᵢ|

B. |pᵣ* · pᵢ|²

C. pᵣ + pᵢ

D. |pᵣ × pᵢ|

Correct Answer: B. |pᵣ* · pᵢ|²

Question 4

In the Friis equation, if received power means available power and transmitted power means accepted antenna power, which factor is not automatically included?

A. Free-space spreading factor

B. Transmit gain

C. Receiver load mismatch

D. Polarization factor when specified

Correct Answer: C. Receiver load mismatch

Question 5

For a small lossless dielectric sphere, Rayleigh scattering cross section varies approximately as:

A. a²λ

B. a³λ⁻¹

C. a⁶λ⁻⁴

D. a⁻⁶λ⁴

Correct Answer: C. a⁶λ⁻⁴

Question 6

The optical theorem relates extinction cross section to:

A. The backward magnetic field only

B. The imaginary part of the forward scattering amplitude

C. The antenna radiation resistance

D. The geometric circumference only

Correct Answer: B. The imaginary part of the forward scattering amplitude

Question 7

Antenna gain is most correctly expressed as:

A. Directivity divided by wavelength

B. Efficiency multiplied by directivity

C. Radiation resistance multiplied by bandwidth

D. Effective aperture divided by frequency

Correct Answer: B. Efficiency multiplied by directivity

Question 8

For two linearly polarized antennas with polarization angle difference ψ, the polarization loss factor is:

A. sinψ

B. cosψ

C. sin²ψ

D. cos²ψ

Correct Answer: D. cos²ψ

Question 9

The far-field approximation for an antenna of maximum dimension D commonly requires:

A. r ≪ D

B. r ≈ D/λ

C. r ≳ 2D²/λ

D. r ≲ λ²/D

Correct Answer: C. r ≳ 2D²/λ

Question 10

For a Hertzian dipole, maximum radiation occurs:

A. Along the dipole axis

B. Broadside to the dipole

C. At every angle equally

D. Only in the reactive near field

Correct Answer: B. Broadside to the dipole