Chapter 5: Worked Examples and Learning Path

Question: which parts of a wavefront design follow from a target formula, and which require a physical scattering calculation?

Use Chapter 5, the browser lab, and the two executed notebooks: beam steering and wavefront models. Their execution order matters: the wavefront notebook reads the expanded library produced by the beam notebook.

Learning path¶

For a classroom demonstration, write six target phases on a circle. First select only from Chapter 3’s partial phase arc, then from the extended library. Compare the resulting geometry sequences before showing either full-wave spectrum. Ask students to predict the identical-cell and reversed-sequence controls.

Example 1: a six-cell beam direction¶

For $p=0.65\,\mu\mathrm m$ and six cells, $L=3.90\,\mu\mathrm m$. At normal incidence, a positive $2\pi$ ramp targets $m=+1$. With $\lambda_0=1.55\,\mu\mathrm m$ and output index 1.5,

Air-side reflection has orders -2 through +2, while glass-side transmission has -3 through +3. Their amplitudes cannot be obtained from this angle calculation. The six-ridge RCWA result supplies those amplitudes.

Example 2: six phase levels and their sinc factor¶

The staircase with $t_j=e^{2\pi i(j+1/2)/6}$ has $C_{+1}=\operatorname{sinc}(1/6)$, so $|C_{+1}|^2\approx0.9119$. This is a scalar coefficient fraction. The finite interval integration supplies the sinc; a point-source sum would omit that interval shape.

Compare this number with the actual supercell’s desired-order power. They refer to different models, and equality is not an acceptance condition. The audit checks the analytical coefficient identity for 2, 3, 4, 6, 8, and 16 levels.

Example 3: expand a library without hiding transmission constraints¶

The coarse height screen, using the original width domain, gives phase spans of approximately 132°, 200°, 271°, 341°, and 412° at heights 0.6, 0.9, 1.2, 1.5, and $1.8\,\mu\mathrm m$. These exploratory curves use $M=16$; the accepted $1.8\,\mu\mathrm m$ library is recalculated and checked at higher order.

A full unwrapped cycle does not guarantee every phase at $T\geq0.8$. The six-target selection must still check each circular error. The saved expanded design has errors below $1^\circ$ at all six positions. It uses a common height, so the transmission reference plane is consistent within the sequence.

Example 4: two denominators for a beam’s performance¶

For the expanded supercell, $T_{+1}\approx0.85536$ and $T_{\rm total}\approx0.91139$. Thus incident-normalized target power is approximately 85.54%, while

The second number excludes reflected incident power. Reporting it as incident efficiency would change the denominator. The local staircase predicts about 80.27% target power, and its reflected-plus-transmitted estimate is about 0.88225. That imbalance is an approximation error, not absorption in this lossless ridge model.

Example 5: identical elements and reversal¶

Six identical ridges preserve the smaller primitive period $p$. In the $6p$ expansion, only orders that are multiples of six can have nonzero amplitudes. At this wavelength none of the nonzero multiples is open, so all open nonzero supercell powers vanish.

For the selected asymmetric sequence, reversing widths mirrors the structure in $x$. At normal incidence the transmitted/reflected order powers exchange $m\leftrightarrow-m$. The strong +1 beam becomes a strong -1 beam. This is a symmetry check independent of whether the design objective was well chosen.

Example 6: a scalar line-focus metric¶

For $D=31.2\,\mu\mathrm m$, $f=30\,\mu\mathrm m$, and output index 1.5, the continuous focusing phase is $-k_t(\sqrt{x^2+f^2}-f)$. The saved scalar FWHM at the design plane is about $0.971\,\mu\mathrm m$ for the continuous aperture, $0.982\,\mu\mathrm m$ for the sampled phase aperture, and $0.985\,\mu\mathrm m$ for the local-library aperture.

The local-library peak squared field is about 0.550 of the continuous reference peak. It includes amplitude reduction and phase/sampling effects. The fraction of modeled line intensity in a fixed window answers another question; it must not be relabeled as a full-wave lens efficiency. The window/grid checks concern propagation of these prescribed fields.

Example 7: a half-wave element at 30 degrees¶

With $t_x=1$, $t_y=-1$, and element rotation $\theta=30^\circ$, a state $\mathbf e_+=(1,i)^T/\sqrt2$ converts to $\mathbf e_-$ with phase $+60^\circ$. The opposite input basis state gives phase $-60^\circ$.

If retardance is only $90^\circ$, the ideal lossless retarder converts a fraction $\sin^2(45^\circ)=1/2$. Geometric-phase control and conversion amplitude require separate checks. These ideal Jones coefficients are not the vector response of our TE ridges.

Report checklist¶

Preserve target fields, phase/reference conventions, illumination, candidate library, selection objective, geometry sequence, local predictions, all-channel full-wave powers, numerical refinements, and control cases. For focusing/polarization, name the scalar or Jones model and its metrics. Retain lower-performing or infeasible cases when they reveal a limitation.

The lab specifies executed stages and further finite-device, independent-method, and optimization outlines. Chapter 6 broadens validation methods; Chapter 10 can reuse versioned simulation data for learned design.