Chapter 2: Self-check and Wave Detective Project

Attempt these questions after the chapter, then open the answers. The worked examples show the intermediate calculations; the lab supplies the browser, notebook, source bundle, and audit. Keep a prediction log: a prediction that fails for a clear reason is useful evidence of learning.

Q1: Which way is the power?¶

A wave in air has $\widetilde{\mathbf E}=E_0\widehat{\mathbf y}e^{ik_0z}$ with time dependence $e^{-i\omega t}$. Which direction must its magnetic field point? What happens when you reverse the propagation direction while keeping electric polarization along $y$?

Q2: Is a large field evidence of gain?¶

An interface from $n_i=1.5$ to $n_t=1$ gives $r=0.2$ and $t=1.2$. A report calls $|t|^2=1.44$ its transmitted power fraction and declares gain. Diagnose the report and give the correct power balance.

Q3: What rotates in a Jones vector?¶

For $(E_x,E_y)^T=(1,i)^T/\sqrt2$ at $z=0$, describe the electric-field trajectory under our time convention. What fraction passes an ideal linear analyzer? Would multiplying the entire Jones vector by one global phase change its polarization ellipse?

Q4: Why is there a minus sign?¶

Derive the two scalar continuity equations for a bare interface using incident tangential electric amplitude one. Why does the magnetic equation contain $1-r$? Which polarization becomes reflectionless at the air-to-glass Brewster angle?

Q5: Where is the transmitted field?¶

Above the glass-to-air critical angle, the transmitted normal power is zero. Does the field vanish everywhere in air? What distinguishes an amplitude decay length from an intensity decay length? What caution applies at exact TM cutoff?

Q6: Why can an extra layer help?¶

An ideal normal-incidence antireflection layer joins $n_0=1$ to $n_2=1.5$. Find its quarter-wave index/thickness at $\lambda_d=0.6\,\mu\mathrm m$. If its reflectance is zero, must the transmission coefficient be 1 with zero phase?

Q7: Is this a valid sheet retrieval?¶

Empty air is described by $r=0,t=1$ at one common plane. Moving only the output reference downstream by $\lambda_0/4$ gives $t=i$. A retrieval $g=2(1-t)/t$ then returns negative conductance. What should you check before declaring an active material? How does fixed absolute amplitude uncertainty affect retrieval near $t=0$?

Q8: Does phase change dissipation?¶

A purely electric equal-air sheet has $g=2$, $r=-1/2$, and $t=1/2$. What is its one-sided absorption? What is absorbed for equal coherent counterpropagating beams of total unit input power and relative phase 0, $90^\circ$, or $180^\circ$? Does this contradict the one-sided absorption bound?

The wave detective project¶

Task: prepare an evidence report resolving the chapter’s three puzzles: a transmitted amplitude above one, a field with no outgoing normal-power channel, and phase-controlled absorption by an unchanged sheet.

1. Predict: sketch the incident/outgoing fields and state your predicted amplitudes, powers, or limits before evaluating them.

2. Reproduce: run the analytical notebook from a fresh kernel. Retain its environment, parameter values, source version, and audit.

3. Challenge: change one quantity at a time: exterior index/angle, film optical thickness, Jones relative phase, and sheet conductance or illumination phase. Include one case where the initial prediction fails.

4. Diagnose: deliberately test one wrong normalization, reference-plane interpretation, or growing evanescent branch on paper. Explain which physical or consistency check exposes the error; preserve the valid model in your submitted computational source.

5. Connect: inspect one saved response in the Chapter 3 library. State its geometry, polarization, indices, wavelength, phase planes, power denominator, and open channels. Explain what the present uniform-interface or local-sheet models do not represent.

6. Propose: write one independent test that would be needed for a specific physical sample. Identify calibration and uncertainty if you propose a measurement.

Submit the executed notebook, at least one parameter record from the browser, the generated figures, a compact prediction/result table, audit output, and a report of about 3–4 pages. Include a failed prediction and its correction. Mark proposed or unexecuted work clearly. Students can share editable Markdown/notebooks through the existing contribution workflow.

Suggested rubric¶

This rubric is a proposed teaching aid for instructor adaptation. A well-explained negative result or rejected model can earn full scientific credit; an attractive figure without the correct denominator cannot replace the reasoning.

A working glossary¶

Continue to Chapter 3 for geometry-dependent amplitudes, Chapter 4 for outgoing diffraction channels, or repeat a doubtful result in the lab before proceeding.