Lab 2: Waves, Interfaces, and a Sheet That Listens to Phase

Primary topic ID: ms-01-waves-interfaces. Additional introductory activities support ms-03-sheet-response and ms-06-polarization; these activity identifiers need not equal chapter numbers.

Question: can you distinguish a field effect, a power transfer, and a change in coherent illumination before you trust a simulation or retrieval?

Read the chapter, use the nine worked examples, and retain a prediction log. The self-check and project provide answer guidance, deliverables, and a suggested rubric.

A. Predict in the browser¶

Status: executable analytical browser exploration. It evaluates equations directly, without a full-wave solver, external dataset, or installation.

Original analytical experiments in wave amplitudes, flux, phase, and response assumptions.

Open the browser lab in a new tab

Use the corresponding experiment controls to investigate:

Amplitude versus power: compare air-to-glass and glass-to-air normal incidence. Predict $r,t,R,T$ and explain why $|t|>1$ is possible without gain.
Angle and polarization: locate the air-to-glass TM Brewster minimum. Compare TE at the same angle, then reverse the index contrast and cross the critical angle.
Evanescence: retain a total-internal-reflection case with nonzero transmitted field coefficient but zero net normal transmitted power. State the decaying branch and distinguish field decay from normal power transfer.
Film interference: compare an uncoated interface with the quarter-wave antireflection design. Change wavelength or thickness while retaining the other parameters.
Polarization: compare a linear state, a circular state, and an ellipse. Predict a linear analyzer’s response. State the axes and time convention before describing rotation.
Sheet response: compare a passive resistive and a reactive electric sheet. Identify the assumptions behind the one-sided 50% absorption bound.

Keep a written parameter/response record, one failed prediction and its explanation, and a short statement of what the browser does not calculate. At exact TM cutoff, use the stated limiting interpretation rather than treating an infinite intermediate admittance as an ordinary field value.

B. Run the analytical notebook¶

Status: executed analytical reference available. The notebook covers the interface, film, polarization, electric-sheet retrieval, and coherent two-sided illumination. It does not run a patterned-geometry Maxwell solver.

Download the complete source: chapter-02-waves.zip. Extract the ZIP before running it, preserving the notebooks/, simulations/, scripts/, and data/ folders. Follow the shared environment instructions. The calculations use ordinary CPU-based numerical libraries; no commercial solver or GPU is required.

Individual files: Chapter 2 notebook: waves, interfaces, and sheets, waves_interfaces.py, waves-reference.json, waves-reference.csv, and validation.json.

Setting	Reference experiment or convention
Time and space	$e^{-i\omega t}$ , $e^{i\mathbf k\cdot\mathbf r}$
Geometry	Interface near $z=0$ ; incidence from $z<0$ ; plane of incidence $xz$
Field basis	Common tangential $E_y$ for TE, $E_x$ for TM
Bulk media	Positive real constant indices, $\mu=\mu_0$
Interface benchmark	$n_i=1$ , $n_t=1.5$ ; reverse for total internal reflection
Film benchmark	Normal incidence, $\lambda_d=0.6\,\mu\mathrm m$ , $n_f=\sqrt{1.5}$ , $d=\lambda_d/(4n_f)$
Sheet	Normal incidence; local purely electric admittance $g=\eta_0Y_s$
Retrieval reference	Equal air media, common sheet plane, $t=1+r$ , nonzero transmission
Polarization	Coherent Jones components and an explicitly stated Cartesian/time convention
Power	Normal flux, with the appropriate admittance and total input denominator

Procedure¶

Write the electric and magnetic signs for forward and backward waves. Derive the bare-interface equations before executing cells.
Restart the kernel and run all cells. Inspect complex amplitudes beside their correctly normalized powers.
Reproduce normal incidence, Brewster cancellation, and a total-internal-reflection case. Check the root branch and treat grazing cutoff separately.
Calculate a quarter-wave film response and change its optical thickness. Retain the entrance/exit phase references.
Inspect the Jones trajectory and analyzer response. Compare global and relative phase changes.
Compare a reactive and resistive sheet. Check surface-current continuity and the power dissipated by its conductance.
Retrieve the known synthetic shunt only under its valid same-plane assumptions. Run the notebook’s controlled retrieval-error experiment: change the fixed complex perturbation’s magnitude/direction, compare exact and linearized parameter errors, and report perturbation magnitude relative to transmission. Its perturbed reflection preserves the model continuity deliberately; independently noisy coefficients need a separate residual check. Identify an incompatible response rather than forcing a fit.
For $g=2$ in air, compare one-sided illumination with normalized equal-power two-sided inputs. Calculate outgoing field sums before powers; sweep relative phase and explain the coherent-absorption result.
Save the executed notebook, generated figures, CSV/JSON, parameter records, and audit output. Preserve the model, source version, and assumptions when modifying an experiment.

What must agree?¶

Check	Interpretation
Air/glass normal incidence	$r=-0.2$ , $t=0.8$ , $R=0.04$ , $T=0.96$
Reverse interface	$r=0.2$ , $t=1.2$ with the same $R,T$
TM Brewster condition	Reflection zero at $\arctan(n_t/n_i)$ in the stated medium model
Total internal reflection	Decay into the second half-space; $R=1$ , $T=0$ without assuming the whole field vanishes
Quarter-wave film	Reflected cancellation and $T=1$ at the ideal design condition
Polarization	Retarder norm preservation, analyzer projection, and stated rotation convention
Passive sheet	$R+T+A=1$ and nonnegative dissipated power
Sheet retrieval	Known synthetic parameter recovered only when the model and phase-plane checks hold
Coherent illumination	Total output calculated from $S\mathbf a$ ; phase-controlled absorption under normalized total input

The saved audit states numerical thresholds and residuals for its test domain. Report these rather than declaring a calculation accurate because a plot looks smooth. Analytical identities and independent formulations check this model; they are not mesh convergence or validation of a patterned device.

C. Connect fields to a geometry¶

Status: a reproducible comparison activity using already published full-wave data. No new ridge simulation or geometry-to-sheet retrieval is performed by this lab.

Choose one accepted response from the Chapter 3 unit-cell library. Record geometry, media, wavelength, polarization, reference planes, complex coefficients, and power fractions. Recalculate its incident-normalized transmitted power with its correct exterior flux factor. Compare that with $|t|^2$ and with the equal-air ideal sheet.

Now ask whether $t=1+r$ at a common sheet plane is even the appropriate model test for that finite-height, substrate-supported structure. Identify the information needed before fitting any sheet parameters. A finite layer’s entrance and exit coefficients cannot be inserted blindly into a zero-thickness shunt inverse.

Deliverable: a one-page evidence comparison connecting a uniform-interface benchmark, a local sheet, and a patterned full-wave cell. Identify a claim that each model can support and an additional test that it cannot replace.

D. An optional physical investigation¶

Status: a proposed later-course extension, not an executed measurement or an additional requirement for completing the introductory lab.

Choose a microwave or optical apparatus suited to one experiment: interface reflection, a thin-film spectrum, polarization analysis, or coherent two-sided absorption. Specify the material/bias model, excitation and spatial mode, phase planes, detector collection, calibration, uncertainty, and safe operating procedure under the laboratory’s own rules. Distinguish a normal-power measurement from a local-field probe. A coherent absorption test also needs controlled relative phase and a measured total input denominator.

Predict the measurable observable before collecting data. Retain a separate condition for checking a fitted model; state finite-aperture, material-dispersion, and alignment limitations. Fabrication, experimental uncertainty, and more general retrieval belong to their detailed later investigations.

Use the wave detective project and rubric, and submit editable sources through the contribution workflow. Include references, environment, source/data versions, figures, conventions, residuals, and execution status. Keep the distinction between an executed analytical result, reused full-wave data, and a proposed experiment visible in the report.