Uniform Plane Waves and Lossy Media Propagation

Unit 3: Boundaries, reflection and dispersive media

What the unit establishes

Apply electromagnetic boundary conditions at interfaces and use energy conservation to verify reflection and transmission. Understand when phase, group delay, and dielectric losses matter to real systems.

Reproducible investigation

Investigate a plane wave encountering a nonmagnetic dielectric interface first at normal incidence, then at oblique incidence with both TE and TM polarizations. Begin from tangential continuity and derive field coefficients rather than copying a reflection chart. Check R+T=1 in a lossless normal-incidence case, and explain the impedance factor in transmitted power. For n1>n2, calculate the critical angle, then sketch the evanescent transmitted field above it. Add a finite dielectric slab: record two interface reflections and the round-trip phase, then vary coating thickness by ten percent and measure which frequency moves toward the reflection minimum. Finally estimate group delay through a dispersive material and explain why a transfer-function phase plot is not a proof about signal-front speed.

Four connected lessons

1. Normal-incidence reflection and transmission

The starting point is the physical situation considered in the [full lesson](../lessons/U03-L01-normal-incidence-reflection-and-transmission.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

For normal incidence from medium 1 to 2, Γ_E=(η2−η1)/(η2+η1) and τ_E=2η2/(η2+η1)=1+Γ_E. For lossless real impedances, R=|Γ_E|² and T=(η1/η2)|τ_E|², giving R+T=1. The transmitted and reflected magnetic field signs depend on wave direction; using the same H sign for forward and reverse waves breaks the boundary condition.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

2. Oblique incidence, Snell law and total internal reflection

The starting point is the physical situation considered in the [full lesson](../lessons/U03-L02-oblique-incidence-snell-law-and-total-internal-reflection.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

For nonmagnetic lossless isotropic media n1 sinθi=n2 sinθt. The critical angle for n1>n2 is θc=arcsin(n2/n1). For TE polarization, one convenient impedance form is Γ_TE=(η2/cosθt−η1/cosθi)/(η2/cosθt+η1/cosθi), with proper attention to effective wave impedance; for TM use η cosθ instead. An easier derivation writes tangential E and H continuity explicitly, avoiding memorization ambiguity.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

3. Thin films, multiple reflections and interference

The starting point is the physical situation considered in the [full lesson](../lessons/U03-L03-thin-films-multiple-reflections-and-interference.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

Round-trip phase in a lossless slab is 2βd where β=2πn/λ0 at normal incidence. A quarter-wave optical thickness means nd=λ0/4, giving round-trip phase π. For a single anti-reflection layer between lossless media at normal incidence, ideal amplitude matching uses n_layer≈sqrt(n_incident n_substrate) with quarter-wave optical thickness. An exact reflectance calculation must sum all multiple reflections or solve a two-port field matrix.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

4. Dispersion, group delay and causal material models

The starting point is the physical situation considered in the [full lesson](../lessons/U03-L04-dispersion-group-delay-and-causal-material-models.html). Its key mathematical relationship is the following, stated with the particular geometrical or material assumptions of this topic:

For a phase factor exp[j(ωt−β(ω)z)], v_phase=ω/β and v_group=dω/dβ=1/(dβ/dω). Over distance L, group delay τg=L dβ/dω. A second-order Taylor expansion β≈β0+β1Δω+(β2/2)(Δω)² quantifies the first-order delay and group-velocity dispersion. A bandwidth-limited pulse experiences differential delay when β2 is nonzero.

Work through its numerical example on paper before using the model. Compare the predicted result with the situation described in the lesson's engineering case; check which assumption restricts the validity of the formula. The goal is not memorization of symbols but identifying whether a simplified scalar parameter is sufficient for this geometry and frequency.

Unit test of understanding

Use the [eight solved questions](../assessments/unit-03.html) as a closed-book audit. Then complete the [unit investigation](../projects/unit-03.html) and compare its computational trend with the [unit interactive model](../simulations/model-03.html). Write down any mismatch between the ideal analytical limit and the plotted values; if an output appears to violate conservation or dimensional consistency, diagnose the implementation before adjusting a physical parameter to force agreement.

Evidence standard

A completed study record contains the source equations, declared time and orientation convention, parameter units, at least two numerical cases, a model-limitation statement and a reproducible test. Nothing in this unit requires live mains or high-voltage apparatus; synthetic numerical work is adequate for the computational project. A credible claim about real equipment would additionally require calibration and qualified verification appropriate to that application.

U03-L01: Normal-incidence reflection and transmission

Book 22 · Unit 3 — Boundaries, reflection and dispersive media Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

At a flat material interface the tangential electric and magnetic fields obey boundary conditions that fix the amplitudes of reflected and transmitted plane waves. Reflection follows from impedance mismatch: if the downstream intrinsic impedance differs from the upstream one, some incident power is returned. A negative electric-field reflection coefficient signifies a phase reversal, not negative energy. A lossless interface must conserve power when transmission coefficients are evaluated with the correct impedance factors. This exercise is an excellent audit because simply squaring an electric amplitude transmission coefficient generally does not give the transmitted power fraction.

Derivation and governing equations

For normal incidence from medium 1 to 2, Γ_E=(η2−η1)/(η2+η1) and τ_E=2η2/(η2+η1)=1+Γ_E. For lossless real impedances, R=|Γ_E|² and T=(η1/η2)|τ_E|², giving R+T=1. The transmitted and reflected magnetic field signs depend on wave direction; using the same H sign for forward and reverse waves breaks the boundary condition.

Fully specified worked example

Vacuum η1≈377 Ω to an ideal nonmagnetic dielectric εr2=4 has η2≈188.5 Ω, Γ_E=(188.5−377)/(565.5)=−1/3. Reflected power fraction is 1/9≈11.1%, transmitted power fraction≈88.9%. The transmitted electric amplitude is 2/3 of the incident amplitude; its power is not (2/3)² of incident power.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

A sensor enclosure may contain several dielectric windows. A single interface reflection is the first benchmark before estimating performance through the full slab. Comparing a dielectric lens design solely by ε_r ignores thickness, angle of incidence, material loss, surface quality and polarization.

Scope, failure modes and falsification

These formulas assume planar infinite interfaces, uniform nonmagnetic isotropic materials and a propagating plane wave. Nearby edges generate diffraction; a finite slab exhibits second-interface reflections. In lossy media the power fractions require the real part of the complex Poynting flux, not naïve real-impedance expressions.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 3 overview](../units/unit-03.html) · [Assessment and answer key](../assessments/unit-03.html) · [Interactive model](../simulations/model-03.html).

U03-L02: Oblique incidence, Snell law and total internal reflection

Book 22 · Unit 3 — Boundaries, reflection and dispersive media Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

An obliquely incident wave conserves the tangential component of wavevector across a planar interface. This phase-matching condition yields refraction and, depending on polarization, different reflection amplitudes. TE and TM waves do not generally share a reflection coefficient. At incidence from higher to lower refractive index, a critical angle may be reached beyond which no propagating transmitted plane wave carries average normal power into the lossless lower-index half-space. A transmitted evanescent field still exists at the interface, so total internal reflection is not an absence of field on the other side.

Derivation and governing equations

For nonmagnetic lossless isotropic media n1 sinθi=n2 sinθt. The critical angle for n1>n2 is θc=arcsin(n2/n1). For TE polarization, one convenient impedance form is Γ_TE=(η2/cosθt−η1/cosθi)/(η2/cosθt+η1/cosθi), with proper attention to effective wave impedance; for TM use η cosθ instead. An easier derivation writes tangential E and H continuity explicitly, avoiding memorization ambiguity.

Fully specified worked example

For n1=1.5 glass-like medium and n2=1.0 air, θc=arcsin(2/3)≈41.81°. At θi=30°, sinθt=1.5×0.5=0.75, so θt≈48.59°. At θi=50°, sinθt would exceed unity, indicating evanescent rather than traveling transmission for an ideal lossless interface.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

In optical fiber, internal ray angles and numerical aperture can aid introductory design, but actual guided modes require boundary-value solutions. Polarization, bend radius, index profile and surface roughness control attenuation and coupling. A ray picture helps initially, yet it is not a substitute for modal analysis near cutoff or in subwavelength structures.

Scope, failure modes and falsification

The refractive-index construction assumes homogeneous isotropic lossless media. Anisotropic crystal refraction, dispersive group directions and absorption need more general propagation rules. At a finite interface an evanescent wave can couple across a thin gap and transfer power to a third medium, as in frustrated total internal reflection.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 3 overview](../units/unit-03.html) · [Assessment and answer key](../assessments/unit-03.html) · [Interactive model](../simulations/model-03.html).

U03-L03: Thin films, multiple reflections and interference

Book 22 · Unit 3 — Boundaries, reflection and dispersive media Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

A finite-thickness slab has two interfaces, so the backward wave from its far face can reflect again from its near face. Their coherent superposition can reinforce or cancel the net reflected amplitude. Film thickness therefore matters through phase delay, not just through material composition. Quarter-wave anti-reflection concepts follow by choosing thickness and refractive index so reflected components approximately cancel at one target wavelength. Outside that band or at different incidence angles the cancellation changes. A multilayer stack is naturally analyzed with a transfer matrix, but its matrix ordering must be documented to avoid swapping left and right sides.

Derivation and governing equations

Round-trip phase in a lossless slab is 2βd where β=2πn/λ0 at normal incidence. A quarter-wave optical thickness means nd=λ0/4, giving round-trip phase π. For a single anti-reflection layer between lossless media at normal incidence, ideal amplitude matching uses n_layer≈sqrt(n_incident n_substrate) with quarter-wave optical thickness. An exact reflectance calculation must sum all multiple reflections or solve a two-port field matrix.

Fully specified worked example

Suppose λ0=600 nm, layer n=1.22, substrate n≈1.5 and ambient n=1. The quarter-wave thickness is d=λ0/(4n)≈123 nm; sqrt(1×1.5)≈1.225 is close to the chosen layer index. A physical coating at that thickness may reduce reflection near 600 nm but is not broadband by construction.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

For an RF radome, the same coherent-slab concept applies at vastly larger wavelengths. The relevant tradeoffs include mechanical thickness, dielectric constant, incidence angle, polarization and absorption. A coating optimized for the nominal center frequency may be poor at the band edge or at large scan angles.

Scope, failure modes and falsification

Perfect cancellation requires appropriate index matching, exact phase and sufficiently small losses. Actual coatings have thickness variations, roughness and dispersion. Incoherent illumination, strong absorption or rough interfaces invalidate the simplest two-wave interference picture.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 3 overview](../units/unit-03.html) · [Assessment and answer key](../assessments/unit-03.html) · [Interactive model](../simulations/model-03.html).

U03-L04: Dispersion, group delay and causal material models

Book 22 · Unit 3 — Boundaries, reflection and dispersive media Learning objective: Reconstruct the physical model, calculate one representative case and explain where its assumptions cease to apply.

Physical principles

Phase velocity describes constant phase; group velocity describes the motion of a narrowband envelope under suitable conditions. In a dispersive material β depends nonlinearly on frequency, so different spectral components accumulate different phase delays. A pulse can broaden even when attenuation is weak. It is essential not to identify any formal group velocity directly with the velocity of information in strongly absorbing or anomalously dispersive regions. Causal material behavior links dispersion and loss; an arbitrary frequency-dependent real permittivity with zero loss may be physically inconsistent over a broad band.

Derivation and governing equations

For a phase factor exp[j(ωt−β(ω)z)], v_phase=ω/β and v_group=dω/dβ=1/(dβ/dω). Over distance L, group delay τg=L dβ/dω. A second-order Taylor expansion β≈β0+β1Δω+(β2/2)(Δω)² quantifies the first-order delay and group-velocity dispersion. A bandwidth-limited pulse experiences differential delay when β2 is nonzero.

Fully specified worked example

If β1=5 ns/m in a guided medium, a 20 m path has group delay 100 ns. If delay varies by 0.2 ns over the signal bandwidth per metre, the 20 m path creates approximately 4 ns of differential group delay. A claim about exact pulse width would additionally require the input waveform and spectral profile.

Independent numerical verification. Reconstruct the arithmetic in the preceding illustration using SI base units, then change one input by a factor of two and predict the consequence from the actual governing expression. Specify which inputs remain fixed. When the result involves a signed voltage or complex amplitude, report the chosen polarity or phase reference; a magnitude alone cannot settle the sign. Compare the result against an analytically known limit rather than blindly trusting significant figures from a calculator.

Engineering decision scenario

Timing links, fiber communication and PCB interconnects all face dispersion but from different sources. A timing engineer should record phase response versus frequency, not only insertion loss. Two links with identical attenuation can yield different bit errors because their group-delay distortion differs.

Scope, failure modes and falsification

Group delay can be poorly behaved near deep transfer-function nulls where phase unwrap is unstable. Material parameters are often complex and temperature-dependent. Negative formal group delay over a narrow band does not establish faster-than-light information transfer.

What evidence would change this conclusion? Propose one measurement or full-wave simulation that would expose the limitation described above, name the raw observable, and determine whether it distinguishes an omitted physical effect from a code implementation error. Numerical convergence and physical validation are different claims. No idealized wave or electromagnetic enclosure model in this chapter constitutes a hardware safety approval.

Three thinking problems

1. Compute and check: Recalculate the central numerical result above with a clearly stated changed parameter. Show unit conversion and the primary scaling law, not just a final decimal.

2. Derive and compare: Starting with the main field law above, reconstruct the next relation using the assumptions explicitly stated in the lesson. Say which boundary condition or sign convention is indispensable.

3. Design critique: Apply the limitation above to the engineering scenario. Specify the experiment, frequency range, geometry or calibration that would need to be checked before an engineer could accept the simple calculation.

Continue: [Unit 3 overview](../units/unit-03.html) · [Assessment and answer key](../assessments/unit-03.html) · [Interactive model](../simulations/model-03.html).

Unit 3 engineering project: Dielectric-window matching design

Type: Computational desk investigation using synthetic data only. No mains wiring or high-voltage hardware work.

The problem and experimental model

Compare a bare air-to-substrate interface with a one-layer quarter-wave coating at the design wavelength. Estimate reflected power using impedance matching and explain the second-surface interference. Then evaluate off-design wavelengths and list the physical changes that a dispersion-aware coating calculator would require.

Required evidence

Submit a target-wavelength calculation, a sensitivity table for thickness errors, and a boundary-condition audit showing R+T=1 in a lossless reference case.

Independent benchmark and failure test

Before generating graphs, derive a baseline from the four unit lessons. Establish a numerical prediction from known dimensions, wave speed or material law. Run a second numerical calculation that uses a different formulation or a limiting case. Inspect whether the model conserves charge or power where the assumptions require it; use the proper amplitude or power reference. Compare input perturbations one at a time so the direction of the observed effect can be explained physically.

Analysis requirements

Every plot must label input units, dependent quantities, coordinate reference and whether the curve is synthetic. A statement such as 'the signal became better' is incomplete without specifying the chosen metric and the control variables. Include a short error budget separating numerical resolution, material uncertainty, geometrical approximation and measurement effects that would matter in a real experiment. An unexplained mismatch must remain visible in the report rather than being edited out.

Deliverables and grading

A concise technical report should include: purpose and assumptions (15%), derivation and units (20%), numerical verification (25%), parameter sensitivity and omitted physics (25%), and evidence-based engineering recommendation (15%). Name one specific datum whose collection would change the recommendation. Do not claim that a synthetic dataset confirms real equipment performance.

References inside the book: [Unit guide](../units/unit-03.html) · [Analytical model](../simulations/model-03.html) · [Assessment](../assessments/unit-03.html).

Unit 3 assessment: Boundaries, reflection and dispersive media

Eight subject-specific questions, with explanatory answers. Attempt all questions before reading the key. These are educational examples, not external exam past papers.

Question 1

For η1=η2 at normal incidence in the lossless plane-wave case, what is reflection coefficient?

Answer and rationale: Γ=0, so the ideal flat interface has no reflection.

Question 2

Why is transmitted power fraction not simply |τE|² for different media?

Answer and rationale: Power includes wave impedance; for lossless real impedances, T=(η1/η2)|τE|².

Question 3

From n1=1.5 to n2=1.0, at what angle does total internal reflection begin?

Answer and rationale: At arcsin(1/1.5)≈41.81° for an ideal lossless interface.

Question 4

Does total internal reflection imply E field is zero beyond the interface?

Answer and rationale: No, an evanescent transmitted field penetrates the lower-index side.

Question 5

For free-space wavelength 500 nm and film n=1.25, what quarter-wave thickness?

Answer and rationale: d=500/(4×1.25)=100 nm.

Question 6

Why can a low-reflection coating fail away from its nominal wavelength?

Answer and rationale: Optical phase thickness changes with frequency; multiple reflected contributions no longer cancel as designed.

Question 7

Can phase velocity exceed c without superluminal information transfer?

Answer and rationale: Yes, formal phase speed need not equal the energy or information-front speed.

Question 8

What measurements help distinguish material dispersion from simple attenuation?

Answer and rationale: Complex frequency response including unwrapped phase and group delay, ideally at multiple lengths or with independent material data.

Beyond the answer key

For at least two numerical problems, write a second route to the result using a limiting case or power/charge conservation argument. For conceptual questions, identify a counterexample to an overgeneralized version of the answer. A problem is not fully understood until the applicable physical assumptions are stated alongside the formula.