# Scientific Sources — Case12

## References

Tao-Yuan Du, Zhong Guan, Xiao-Xin Zhou, and Xue-Bin Bian. 2016. *Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields*. *Physical Review A*, 94, 023419. [Source](https://doi.org/10.1103/PhysRevA.94.023419)

Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields

## Criterion-to-source crosswalk

Article page identifiers below are journal pages, not PDF viewer indices. Supplemental pages are explicitly marked. The crosswalk distinguishes direct source facts, mathematical entailments and task-defined evidence duties. Original images and PDFs are referenced only; none are reproduced.

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1, F2 | Du et al. (2016), article pp.023419-1-2, Eqs.(1)-(7), paragraph choosing the k=0 valence-top state: specified periodic SAE model and xE+epsilon*x^2*E interaction. Inhomogeneity is the stated interaction, not a separately substituted force law. |
| F3 | Du et al. (2016), article p.023419-2, Eq.(8) and following Fourier paragraph: current-derived harmonic spectrum. Equivalent radiation forms require consistent frequency/boundary mapping. |
| O1, O2, I1, M1, M2 | Du et al. (2016), article p.023419-2, Fig.2(a-c), caption and two-plateau paragraph: matched epsilon=0,0.0002,0.0005 total spectra and selective high-energy enhancement rather than common scaling. |
| O3, I2, M3, M4, M5 | Du et al. (2016), article pp.023419-3-4, Fig.3, Fig.5(c,d), Fig.6 and transition-probability paragraphs: small lower-band versus much larger higher-band change and coupling to the valence state. The article does not define a unique projected-current cross-term allocation; task accepts functionally equivalent causal evidence. |
| O4, I3, M6 | Du et al. (2016), article p.023419-3, Fig.4(a,b) full response and Fig.4(c,d) interband illustrations: inhomogeneity changes emission/pathway structure. The caption does not assign the nonzero map an unambiguous epsilon from the two scan values; grade a candidate-declared case, not guessed panel identity. |
| M7 | Du et al. (2016), article pp.023419-2-3, Fig.2(b,c) odd/even spectra, broken-symmetry paragraph and Fig.4 time-frequency discussion: pathway changes and parity-permitted supporting structure. |
| M8, V1, V2 | Du et al. (2016), article pp.023419-1,4-5, one-dimensional SAE model and summary; bounded selectivity and triggered numerical/physical-alternative treatment are task obligations, not absolute-yield or universal experimental-efficiency requirements. |

## Formula and convention notes

For F2-F3/M3-M5, article pp.023419-1-2, Eqs.(1)-(3),(8),
$$H(t)=\frac{p^2}{2}-0.37[1+\cos(2\pi x/8)]+E(t)(x+\epsilon x^2),\quad j(t)=-\langle p\rangle.$$
If acceleration is used, Ehrenfest evolution for this stated Hamiltonian gives $\dot{\langle p\rangle}=-\langle V'(x)\rangle-E(t)[1+2\epsilon\langle x\rangle]$ (subject to any separately accounted boundary/mask effects). Substituting $1+\epsilon x$ as the force while retaining this Hamiltonian is not equivalent. Band population evidence may use $n_b(t)=\langle\psi(t)|P_b|\psi(t)\rangle$ with declared projectors. Source Fig.6 compares $|\langle\varphi_{n_0}|x|\varphi_n\rangle|^2$ and $|\langle\varphi_{n_0}|x^2|\varphi_n\rangle|^2$; these illustrate a mechanism, not a mandatory decomposition or permission to add component spectral powers.

## Scope of the source anchors

The locators identify scientific objects and mechanisms, not compulsory figure layouts, solver settings, exact grids, pointwise tolerances or a fixed evidence package. Canonical examples may be replaced by functionally equivalent evidence subject to the rubric. Source readback verifies the candidate's science; it cannot supply missing candidate presentation, identification or inference. Reporting, applicability/N/A and descriptive Research Closure duties remain those stated in the rubric and task.