AI 中文总结
研究TE和TM波在衍射光栅上的散射问题，利用基于量子动力学思想的动态公式，推导出超出傍轴近似的衍射光束振幅解析表达式，给出了TE和TM波入射到Berry光栅推广形式上的散射问题精确解。
AI 中文摘要
1998年，Berry给出了一个解析解，用于确定原子入射到由势函数 $v(x,y):=iV_0(e^{i\mathfrak{K}\,y}-1)$（$0\leq x\leq\ell$，其他区域为0）描述的光栅上时的衍射光束强度，该结果依赖傍轴近似，适用于由相对介电常数 $\hat\varepsilon(x,y):=1 - v(x,y)/k^2$ 描述的非磁性光栅对横向电（TE）波的衍射。本文表明Berry光栅属于一类更大的可能有磁性的衍射光栅，其对TE和横向磁（TM）波的散射问题都可精确求解。利用基于将散射问题映射到由有效非厄米哈密顿算符产生的量子动力学思想的动态公式，推导出了超出傍轴近似有效的衍射光束振幅的显式解析表达式，并给出了TE和TM波入射到Berry光栅推广形式上的散射问题的精确解。
英文摘要
In J.~Phys.~A {\bf 31}, 3493 (1998), Berry provides an analytic solution of the problem of determining the diffracted beam intensities for atoms incident upon a grating given by the potential, $v(x,y):=iV_0(e^{i\mathfrak{K}\,y}-1)$ for $0\leq x\leq\ell$ and $v(x,y):=0$ otherwise, where $V_0, \mathfrak{K}$, and $\ell$ are positive real parameters. This result, which relies on the paraxial approximation, readily applies to the diffraction of transverse electric (TE) waves by the nonmagnetic optical grating given by the relative permittivity, $\hat\varepsilon(x,y):=1-v(x,y)/k^2$, where $k$ is the incident wavenumber. We show that Berry's grating belongs to a larger class of possibly magnetic diffraction gratings whose scattering problem for both TE and transverse magnetic (TM) waves are exactly solvable. Our analysis makes use of a recently developed dynamical formulation of stationary scattering that is based on the idea of mapping the scattering problem to the quantum dynamics generated by an effective non-Hermitian Hamiltonian operator. {We use this formulation to derive explicit analytic expressions for the diffracted beam amplitudes that are valid beyond paraxial approximation.} As a concrete example, we offer the exact solution of the scattering problem for TE and TM waves incident upon a generalization of Berry's grating.
Comments34 pages, 4 figures, typos corrected, notation improved
