EE7500 Advanced topics in RF & Photonics, Jan - May 2026

1. Maxwell’s Prediction and Hertz’s Verification (1865-1888)

James Clerk Maxwell unified electricity and magnetism in his 1865 equations, predicting electromagnetic waves travel at the speed of light. Heinrich Hertz experimentally verified this in 1888, generating and detecting radio waves for the first time.

Research: How did Hertz’s spark gap transmitter and loop antenna detector work? What was the wavelength of his first transmitted waves?

2. Polarization in Modern Communication Systems

Polarization diversity is exploited in modern systems:

Investigate: Why does GPS use RHCP? How do polarization-maintaining optical fibers work?

3. Millimeter Wave Propagation and 5G Challenges

The high attenuation you calculated (4.8 cm skin depth at 10 GHz) becomes critical for mmWave 5G (24-100 GHz). Rain attenuation at 60 GHz can exceed 15 dB/km, and oxygen absorption creates a peak at 60 GHz.

Explore: Why is the 60 GHz band both a challenge and opportunity? How do massive MIMO and beamforming overcome mmWave propagation losses?

4. Atmospheric Propagation Windows

The atmosphere has "windows" where EM waves propagate with minimal attenuation, determined by molecular absorption. Research the propagation characteristics at: VLF (submarine communication), HF (ionospheric reflection), microwave (satellite links), and optical frequencies. Why can’t we use 22 GHz or 183 GHz for satellite communication?

5. Multipath Fading and Diversity Techniques

Multipath propagation causes destructive interference (Rayleigh fading in urban environments). Modern systems use:

Study: How does the coherence bandwidth relate to delay spread? Why does urban mobile communication (~1-2 GHz) experience more severe multipath than rural?

1. Oliver Heaviside and the Telegrapher’s Equations (1880s)

Oliver Heaviside developed the telegrapher’s equations to analyze signal transmission on telegraph lines, introducing the concept of impedance and developing operational calculus (precursor to Laplace transforms). His work enabled long-distance telephony by showing that adding inductance (loading coils) could reduce distortion.

Research: What were "Pupin coils" and how did they revolutionize telephone networks? Why was Heaviside’s work initially rejected by the scientific establishment?

2. Transmission Lines in Modern High-Speed Digital Systems

Transmission line effects dominate when signal rise times are comparable to the propagation delay. For a 10 cm PCB trace with ε_r = 4.5, propagation delay is ~1 ns. Rise times in modern systems:

Investigate: What are the key differences between electrical length and physical length? How do vias and connectors affect signal integrity?

3. Coaxial Cables vs. Microstrip Lines

Two fundamental transmission line types with different trade-offs:

Coaxial cables:

Microstrip lines:

Explore: Why do cell phone PCBs use microstrip at GHz frequencies? What is the "critical frequency" where a trace becomes a transmission line?

4. Smith Chart: The Analog Computer for RF Engineers

Invented by Phillip Smith in 1939, the Smith chart is a graphical tool for solving transmission line problems without calculators. It represents complex impedances on the complex reflection coefficient plane.

Study: How does the Smith chart represent impedance matching? Why is it still used despite modern computational tools? Practice: Plot the impedance transformation for the quarter-wave example in Question 5.

5. Time-Domain Reflectometry (TDR)

TDR sends a fast pulse down a transmission line and observes reflections to diagnose faults:

Modern applications:

Research: How does a TDR distinguish between an open circuit at 10 m and a capacitive discontinuity at 10 m? What is the relationship between TDR and the S₁₁ measurement on a vector network analyzer?

1. The Wilkinson Power Divider (1960)

Ernest Wilkinson invented the resistive power divider in 1960, providing a simple way to split RF power equally between two ports with:

The key innovation: a quarter-wave transformer on each arm with a resistor between outputs. This elegant design is still ubiquitous in RF systems.

Research: How does the isolation resistor work? Why is the Wilkinson divider preferred over a simple T-junction? Design a 3-way Wilkinson divider.

2. Broadband Matching: From Fano to Modern UWB Systems

Robert Fano (1950) proved fundamental limits on matching bandwidth: you cannot achieve perfect match over infinite bandwidth with finite-complexity networks. The gain-bandwidth product is limited by the load Q-factor.

Modern broadband techniques:

Investigate: What is the Bode-Fano limit? How do ultra-wideband (UWB) antennas achieve 3-10 GHz bandwidth?

3. Tunable Matching Networks in Software-Defined Radio

Modern SDRs operate across wide frequency ranges (100 MHz - 6 GHz) and need dynamic impedance matching. Solutions include:

Key challenge: Tuning speed vs. power handling vs. insertion loss

Explore: How does the iPhone antenna tuner work across 30+ cellular bands? What are the trade-offs between electromechanical vs. solid-state tuning?

4. Impedance Matching in Biomedical Applications

Critical for wireless power transfer (WPT) to implantable devices:

Challenge: Human tissue is lossy (high \(\varepsilon_r = 50\text{-}80\), conductivity \(\sigma = 0.5\text{-}2\) S/m), and the "load" impedance varies with patient movement, hydration, and tissue type.

Study: How is impedance matching achieved when the load changes dynamically? What frequencies are optimal for in-body power transfer?

5. Vector Network Analyzers and S-Parameters

Modern VNAs measure scattering parameters (S-parameters) to characterize RF components:

The VNA contains:

Research: What is de-embedding? How does SOLT (Short-Open-Load-Thru) calibration remove cable and fixture effects? Why are S-parameters preferred over Z-parameters at RF?

1. The Discovery of Skin Effect (1883-1886)

Horace Lamb (1883) and Oliver Heaviside (1886) independently discovered that alternating current concentrates near the surface of conductors. This "skin effect" was initially a curiosity but became critical for:

Research: What is "proximity effect" and how does it differ from skin effect? How did early RF engineers compensate for skin effect before understanding the physics?

2. Litz Wire and High-Frequency Conductors

To mitigate skin effect losses, engineers developed specialized conductors:

Litz wire: Many thin, individually insulated strands twisted together

Hollow conductors: Since current flows only in skin depth, why use solid wire?

Silver plating: Silver has σ = 6.3 × 10⁷ S/m (9% better than copper)

Investigate: At what frequency does a 1 mm copper wire have 50% more loss than DC? How thick should silver plating be at 10 GHz?

3. Electromagnetic Shielding and EMC

The IBC simplifies shielding effectiveness (SE) calculations:

For a good conductor of thickness t:

Plus reflection loss at the boundaries.

Modern challenges:

Explore: What is the "shielding effectiveness" of a 0.1 mm aluminum foil at 2.4 GHz? Why do microwave oven doors have perforated metal screens?

4. Frequency-Selective Surfaces: Engineering Boundary Conditions

Frequency-selective surfaces (FSS) are periodic arrays of metallic patches or apertures that act as spatial filters, exploiting boundary conditions to control electromagnetic wave transmission and reflection.

Basic principle:

Unlike uniform conductors which reflect all frequencies, FSS can be designed to:

Common FSS geometries:

Real-world applications:

Radome design for aircraft:

Architectural electromagnetic shielding:

Stealth technology:

Antenna reflectors:

Explore: How is FSS periodicity related to operating wavelength? What happens when the angle of incidence changes? Why are fractal FSS geometries (Sierpinski, Koch) advantageous?

5. Superconducting RF Cavities

Below critical temperature T_c, superconductors have zero DC resistance, but AC losses persist:

Normal conductors at RF:

Superconductors (niobium, T_c = 9.2 K):

Challenges:

Study: Why does RF resistance in superconductors increase as f² instead of √f? What limits the maximum accelerating gradient in SRF cavities?

1. George Green and the Birth of Potential Theory (1828)

George Green, a self-taught miller’s son, published "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism" in 1828, introducing:

His work went unnoticed for 20 years until William Thomson (Lord Kelvin) discovered it. Green died young at 49, never knowing his profound impact.

Research: What was Green’s original physical insight about potential functions? How did his miller occupation influence his mathematical thinking about energy?

2. Machine Learning and Computational Electromagnetics

Recent trend: replacing expensive EM simulations with ML models.

Surrogate modeling:

Inverse design:

Physics-informed neural networks (PINNs):

Challenges:

Research: What is "adjoint variable method" in EM optimization? How does it compare to ML approaches? Can generative AI (diffusion models) design antennas?

Future directions:

1. Line Sources and Cylindrical Waves: The Physical Reality of 1D Green’s Functions

While the 1D Green’s function may seem like a mathematical abstraction, it describes real physical systems involving infinite line sources—sources that extend indefinitely along one direction.

Physical examples of line sources:

Cylindrical wave propagation:

Unlike the 1D Green’s function (constant amplitude), a true infinite line source in 3D space radiates cylindrical waves:

where \(\rho\) is radial distance from the line and \(H_0^{(2)}\) is the Hankel function of second kind.

Relationship to 1D case:

The 1D Green’s function \(g(x,x') = -j/(2k_0) \exp(-jk_0|x-x'|)\) describes propagation along the line, while cylindrical waves describe radiation away from the line. Both are needed for a complete description.

Practical applications:

Historical note:

Arnold Sommerfeld’s 1909 solution for a vertical dipole above a lossy ground plane required evaluating the Sommerfeld integral—essentially integrating the 1D Green’s function over all possible plane-wave components. This problem remained challenging for decades and motivated development of asymptotic methods.

Research: What is the "Norton surface wave" for vertical antennas over ground? How does it differ from the direct radiation? At what distance does the \(1/\rho\) cylindrical spreading transition to \(1/r^2\) spherical spreading for finite-length line sources?

2. Leaky-Wave Antennas: 1D Green’s Functions in Action

Leaky-wave antennas exploit the 1D Green’s function physics—a guided wave that continuously "leaks" radiation along its length, behaving like a continuous line source.

Basic principle:

Historical development:

Modern applications:

Why 1D Green’s function matters:

The radiation pattern of a leaky-wave antenna is computed by integrating the 1D Green’s function (the response to a point source) along the antenna length, weighted by the leakage rate α(z):

This convolution determines the beam pattern and scanning properties.

Explore: What is a composite right/left-handed (CRLH) leaky-wave antenna? How does it achieve backward-to-forward beam scanning? Why are metamaterials useful for leaky-wave designs?

3. The Fourier Transform and the Birth of Signal Processing

The Fourier transform technique used to derive the 1D Green’s function has an epic history connecting pure mathematics, physics, and engineering.

Historical timeline:

The controversy:

Fourier’s claim that "any" function could be represented by sines and cosines seemed absurd to contemporaries (Lagrange, Laplace). The notion of convergence for discontinuous functions wasn’t rigorous until 50 years later.

Modern perspective:

The Fourier transform is actually a unitary operator on L²(ℝ)—it’s an isomorphism between time and frequency domains. Parseval’s theorem shows it preserves energy, making it fundamental to:

Why the pole treatment matters:

The "low-loss medium" trick (k₀ → k₀ - jα) elegantly enforces causality through analytic continuation. This connects to:

Research: What is the "uncertainty principle" for Fourier transforms? How does it relate to Heisenberg’s quantum uncertainty? What are "distributions" (generalized functions) and why did Laurent Schwartz win the Fields Medal for formalizing them?

4. Method of Moments: Numerical Green’s Functions

When analytical Green’s functions are unavailable (complex geometries, inhomogeneous media), numerical methods use the same conceptual framework.

Method of Moments (MoM):

Roger Harrington’s 1968 book "Field Computation by Moment Methods" established the standard approach:

Key advantages:

Modern developments:

Applications:

Explore: What is the "fast multipole method"? How does it achieve O(N log N) complexity? What’s the difference between MoM and finite element method (FEM)?

1. Bessel and Hankel: The Mathematics of Cylindrical Waves

Friedrich Bessel (1784-1846) first derived Bessel functions in 1817 while studying planetary perturbations—specifically, the motion of planets under gravitational influence. He had no idea his functions would become fundamental to electromagnetic wave theory a century later.

Historical development:

Why Hankel functions matter for electromagnetics:

Hankel saw that \(J_\alpha(x)\) and \(Y_\alpha(x)\) (real functions) represent standing waves, while their complex combinations \(H_\alpha^{(1)} = J_\alpha + jY_\alpha\) and \(H_\alpha^{(2)} = J_\alpha - jY_\alpha\) naturally represent traveling waves—perfect for radiation problems.

This was a profound insight: the same way \(e^{jkx}\) is more natural than \(\cos(kx)\) and \(\sin(kx)\) for traveling waves in 1D, Hankel functions are more natural than Bessel functions for cylindrical waves.

Modern applications:

Computational challenges:

Bessel and Hankel functions are transcendental—they cannot be expressed in terms of elementary functions. For large arguments (\(x \gg 1\)), asymptotic expansions work well. For small arguments (\(x \approx 0\)), series expansions are used. Modern libraries (MATLAB, Python scipy.special) use sophisticated algorithms combining both approaches.

Research: What is the "Bessel function of the first kind of fractional order"? How are Bessel functions computed numerically (recurrence relations, continued fractions)? What is the connection between Bessel functions and Fourier-Bessel series?

2. Arnold Sommerfeld and the Radiation Condition (1912)

Arnold Sommerfeld (1868-1951) was one of the giants of theoretical physics, contributing to quantum mechanics, atomic theory, and electromagnetic wave propagation. His radiation boundary condition, introduced in 1912, solved a fundamental problem in electromagnetic theory.

The problem:

When solving wave equations in unbounded domains (infinite space), the differential equation alone has infinitely many solutions. How do we select the physically correct one representing radiation (outward-propagating waves)?

Sommerfeld’s insight:

For a source at the origin radiating in 3D, as \(r \to \infty\), the field must behave like an outward-propagating spherical wave:

+

This implies:

+

This condition uniquely selects the outgoing wave solution and eliminates non-physical incoming waves from infinity.

Extension to 2D:

For cylindrical waves, Sommerfeld showed the condition becomes:

+

with the \(\sqrt{r}\) factor reflecting cylindrical (not spherical) spreading.

Impact:

Sommerfeld’s radiation condition became the foundation for: - Antenna theory (Schelkunoff, Silver, Balanis) - Scattering theory (Mie scattering, radar cross-sections) - Perfectly Matched Layers (PML) in computational electromagnetics - Absorbing boundary conditions in FDTD simulations

Sommerfeld’s legacy:

Beyond the radiation condition, Sommerfeld supervised or influenced an extraordinary group of physicists: Werner Heisenberg, Wolfgang Pauli, Peter Debye, Hans Bethe, and many others. His 1949 lecture notes "Partial Differential Equations in Physics" remain a classic.

Explore: What is the "Silver-Müller radiation condition" and how does it differ from Sommerfeld’s? How are radiation boundary conditions implemented numerically in finite element or finite difference methods?

3. Cylindrical Antennas and the 2D Green’s Function

The 2D Green’s function \(g = -(j/4)H_0^{(2)}(kr)\) describes cylindrical wave propagation—the fundamental building block for analyzing cylindrical antennas and arrays.

Infinite wire antennas:

An infinitely long wire carrying uniform current \(I\) creates a field:

+

This is exactly the 2D Green’s function multiplied by the source strength \(-j\omega\mu I\).

Finite cylindrical antennas:

Real antennas have finite length \(L\). The field is obtained by integrating the 2D Green’s function over the antenna length with the current distribution \(I(z')\):

+

where \(\rho = \sqrt{x^2 + y^2}\) is the radial distance from the antenna axis.

Applications:

Cylindrical array antennas:

Arranging elements on a cylinder (rather than a line or plane) enables: - 360° azimuthal coverage - Electronic beam steering in azimuth - Applications: 5G base stations, radar, direction finding

The array pattern involves summing 2D Green’s functions from each element position.

Computational methods:

Research: What is the "near field" vs. "far field" transition for cylindrical waves? How does the Hankel function asymptotic form connect to far-field antenna patterns?

4. Special Functions in Computational Electromagnetics

Bessel and Hankel functions are just the tip of the iceberg. Electromagnetic problems across different geometries require various special functions:

Coordinate systems and their special functions:

Why so many functions?

Each special function arises from separation of variables in the Helmholtz equation \(\nabla^2\psi + k^2\psi = 0\) in a particular coordinate system. The boundary conditions and geometry determine which function set is natural.

Numerical computation challenges:

Modern libraries (MATLAB besselj, Python scipy.special.jv) use Miller’s algorithm (backward recurrence) and other sophisticated techniques.

Software tools:

Machine learning for special functions:

Recent research explores using neural networks to approximate special functions: - Faster evaluation than traditional methods - Difficult for exotic parameter ranges or high precision - Active area of research

Study: What is "catastrophic cancellation" in computing Bessel functions? How does Miller’s backward recurrence algorithm work? What are "Bessel function zeros" and why are they important for waveguide problems?

5. From 2D to 3D: The Physics of Dimensional Crossover

The transition from 2D to 3D wave propagation reveals fundamental physics about how dimension affects wave behavior.

Power conservation and spreading:

Consider a source radiating power \(P\):

Dimensional crossover in real antennas:

A finite linear antenna of length \(L\) exhibits dimensional crossover:

The transition occurs around \(r \sim L\).

Applications:

Fresnel vs. Fraunhofer regions:

The transition reflects dimensional crossover from extended source to point source.

Metamaterials and effective dimensionality:

Some metamaterial structures confine waves to effectively lower dimensions: - Surface plasmons: 3D light → 2D surface waves - Photonic crystals: Create effective 1D or 2D wave propagation in 3D structures

Quantum analogs:

Similar dimensional effects appear in quantum systems: - 2D electron gas in semiconductors (quantum wells) - 1D quantum wires - 0D quantum dots - Different density of states: \(\rho(E) \propto E^{(d-2)/2}\) where \(d\) is dimension

Research: What is the "radiation resistance" of a dipole antenna and how does it change with length? How do 2D materials like graphene support electromagnetic surface waves? What is the connection between the Green’s function singularity and the dimension of space?