本章把 Bloch 定理从晶体周期势 V(r) 推广到光学周期介质 ε(z)。证明:在折射率周期排布的多层薄膜(H/L/H/L/...)中存在频率区间,电磁波无法以传播模式存在——这就是光子能带间隙(Photonic Bandgap, PBG)。涂膜彩色化的全部物理学,都还原到这一个机制:把 PBG 中心频率调到对应的可见光波长。
This chapter extends Bloch's theorem from crystal periodic potential V(r) to optical periodic media ε(z). The proof: in periodic dielectric stacks (H/L/H/L/...), certain frequency intervals admit no propagating EM modes — the photonic bandgap (PBG). All coating-based colorization reduces to this single mechanism: tuning the PBG center frequency to the target visible wavelength.
由此推出 Bragg 反射镜的两个工程公式:中心频率 ω₀ = πc/(n_H·d_H + n_L·d_L),相对带宽 Δω/ω₀ = (4/π)·arcsin[(n_H−n_L)/(n_H+n_L)]。这是涂膜设计最干净的"配方"——对于 545 nm 翠绿色,TiO₂ (2.45) + SiO₂ (1.46),单层 55.6 / 93.3 nm,7 层堆栈即可获得 R>47% 的反射峰,FWHM 80 nm,NIR 反射率天然<6%。
From this follow two engineering formulas for Bragg mirrors: center frequency ω₀ = πc/(n_H·d_H + n_L·d_L); relative bandwidth Δω/ω₀ = (4/π)·arcsin[(n_H−n_L)/(n_H+n_L)]. The cleanest "recipe" for coating: for 545 nm emerald, TiO₂ (2.45) + SiO₂ (1.46), thicknesses 55.6 / 93.3 nm, a 7-layer stack delivers R>47% peak with FWHM 80 nm and intrinsic NIR reflectance <6%.
实际工程化设计无解析解,由 TMM 主方程(特征矩阵 M_j → 总矩阵 M → 反射率 R(λ))+ 差分进化算法(种群 200 × 1000 代)数值求解。多目标损失 L = w₁·‖R−R*‖² + w₂·(1−η_ret) + w₃·ΔE,权重 (0.5, 1.5, 1.0)——w₂ 取最大不是巧合:把"保发电"放到最高优先级,是 92.3% 超越行业 78% 的核心工程决策。
Real engineering has no closed form — solved numerically via the TMM master equation (per-layer M_j → total M → reflectance R(λ)) + Differential Evolution (population 200 × 1000 generations). Multi-objective loss L = w₁·‖R−R*‖² + w₂·(1−η_ret) + w₃·ΔE with weights (0.5, 1.5, 1.0) — the maximal w₂ is no accident: prioritizing power preservation is the core engineering decision behind 92.3% beating the 78% industry mean.
涂膜 vs 染料 — 机制级差异,不是工艺差距:染料是分子能级吸收(减法着色),分子振动展宽不可避免地把硅敏感的 NIR 也吃掉,电效率损失 25-40%;涂膜是相干干涉反射(加法着色),可显式约束 NIR 反射率<6%,电效率损失 5-12%。把染料工艺优化到极致仍突破不了 75% 上限——这是物理决定的天花板,不是工艺水平问题。
Coating vs dye — mechanism-level, not craft-level: dyes work by molecular electronic absorption (subtractive), with vibrational broadening inevitably eating into silicon's NIR-sensitive band — 25-40% efficiency loss. Coatings work by coherent interference reflection (additive), with explicit constraint of NIR reflectance below 6% — 5-12% efficiency loss. Optimizing dye craft to perfection cannot break the 75% ceiling — that's a physics-imposed wall, not a craftsmanship issue.
论文进一步在 4.1.6 节梳理了三种嵌入纳米结构的扩展机制:(1) 嵌入半导体量子点 → Brus 公式给出尺寸调谐带隙;(2) 嵌入金属纳米颗粒 → LSPR 共振 ω_LSPR ≈ ω_p/√(1+2ε_m);(3) 梯度折射率 GRIN → Bruggeman 等效介质理论。这三种机制可与 Bragg 基底叠加,构成"基础堆栈 + 嵌入 QD + 嵌入 Au + GRIN 渐变"的复合涂膜——这是论文 8.4 节"未来研究方向 1:多机制协同"的物理基础。
Section 4.1.6 further develops three embedded nanostructure mechanisms: (1) semiconductor QD embedding → Brus formula for size-tuned gap; (2) metallic NP embedding → LSPR resonance ω_LSPR ≈ ω_p/√(1+2ε_m); (3) graded-index GRIN → Bruggeman effective medium theory. All three layer onto a Bragg base, forming "stack + QD + Au NP + GRIN" composite coatings — the physics underpinning Section 8.4 future direction 1: multi-mechanism synergy.