Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law

[Nonzero tangential traces] For each $j$, the six-component tangential trace datum $A_j:=(e_j^{\tau },h_j^{\tau })\in {\mathbb{C}}^6$ is nonzero. Suppose $A_j=0$; then $e_j^{\tau }=0$ and $h_j^{\tau }=0$, so $e_j=an$ with $a=e_j\cdot n$ and $h_j=bn$ with $b=h_j\cdot n$ by the orthogonal decomposition of [F3]. Taking the inner product of the amplitude condition $k_j\times e_j={\omega }_j\mu h_j$ of [F1] with $n$ gives $0={\omega }_j\mu b$: indeed $⟨k_j\times e_j,n⟩=\det [k_j,e_j,n]=a\det [k_j,n,n]=0$ because a determinant with two equal columns vanishes and $⟨h_j,n⟩=b$; hence $b=0$ and $h_j=0$. Then the amplitude condition $k_j\times h_j=-{\omega }_j\epsilon e_j$ gives $0=-{\omega }_j\epsilon an$, so $a=0$, and then $e_j=an=0$, contradicting $e_j\mathrm{\ne }0$ in [F1].

[Interface traces as characters] For a wave $j$ and $x=(y,0)$ on the interface, $k_j\cdot x={\kappa }_j\cdot y$ because $k_j\cdot n$ multiplies the vanishing third coordinate; hence at the interface the tangential trace is $E_j^{\tau }(y,0,t)=\mathrm{Re}\left(e_j^{\tau }{e}^{i({\kappa }_j\cdot y-{\omega }_{j}t)}\right)$ and $H_j(y,0,t)=\mathrm{Re}\left(h_j{e}^{i({\kappa }_j\cdot y-{\omega }_{j}t)}\right)$, whose normal parts are annihilated by $n\times \cdot$. The two tangential interface identities of [A1] are identities of tangential vector fields, and for a tangential $w$ the vanishing $n\times w=0$ forces $w=0$ (in the frame $n=e_3$, $w=(w_1,w_2,0)$ gives $n\times w=(-w_2,w_1,0)$ by [F3]); hence they say, for all $z=(y,t)\in {\mathbb{R}}^2\times I$, that $\mathrm{Re}\left(A_t{e}^{i{\lambda }_t\cdot z}\right)=\mathrm{Re}\left(A_i{e}^{i{\lambda }_i\cdot z}\right)+\mathrm{Re}\left(A_r{e}^{i{\lambda }_r\cdot z}\right)$ componentwise, with ${\lambda }_j:=({\kappa }_j,-{\omega }_j)\in {\mathbb{C}}^3$ and $A_j=(e_j^{\tau },h_j^{\tau })$; equivalently $\mathrm{Re}(A_i{e}^{i{\lambda }_i\cdot z}+A_r{e}^{i{\lambda }_r\cdot z}-A_t{e}^{i{\lambda }_t\cdot z})=0$ on the nonempty open set ${\mathbb{R}}^2\times I$, by the additivity of the real part in [F4] and the componentwise linearity of the tangential projection in [F3].

[Frequency grouping] Group the three frequencies of step 2.1 by value: let ${\nu }_1,\dots ,{\nu }_M$ be the distinct values among ${\lambda }_i,{\lambda }_r,{\lambda }_t$ and put $c_{{\nu }_0}:={\sum }_{j:{\lambda }_j={\nu }_0}{ϵ}_j{A}_j$ where ${ϵ}_i={ϵ}_r:=1$ and ${ϵ}_t:=-1$. Then $\mathrm{Re}\left({\sum }_{k=1}^M{c}_{{\nu }_k}{e}^{i{\nu }_k\cdot z}\right)=0$ on ${\mathbb{R}}^2\times I$, the ${\nu }_k$ being pairwise distinct, so [F2] applied to the distinct frequencies ${\nu }_1,\dots ,{\nu }_M$ with coefficients $c_{{\nu }_1},\dots ,c_{{\nu }_M}$ gives $c_{\nu }+{\sum }_{k:{\nu }_k=-\bar{\nu }}\overline{c_{{\nu }_k}}=0$ for every $\nu \in {\mathbb{C}}^3$. Fix $k$ and take $\nu ={\nu }_k$: the left sum is $c_{{\nu }_k}$ alone, while $-\overline{{\nu }_k}=(-\overline{{\kappa }_k},+{\omega }_k)$ has last coordinate ${\omega }_k>0$, whereas every ${\nu }_l={\lambda }_j$ for some $j$ has last coordinate $-{\omega }_j<0$; hence ${\nu }_l=-\overline{{\nu }_k}$ fails for every $l$, the second sum is empty, and $c_{{\nu }_k}=0$.

[All frequencies agree] By step 3.1, for each distinct actual frequency $\nu$ the signed sum of the tangential data of the waves with that frequency vanishes. If the three actual frequencies were not all equal, then some actual frequency would occur exactly once among ${\lambda }_i,{\lambda }_r,{\lambda }_t$ (a partition of three terms into at least two nonempty groups has a group of size one), and for that $\nu$ the identity $c_{\nu }=0$ of step 3.1 would read $\pm A_j=0$ for the corresponding single wave, contradicting the nonzero trace datum of step 1.1. Hence ${\lambda }_i={\lambda }_r={\lambda }_t$, which is the assertion (a): reading the three coordinates, ${\kappa }_i={\kappa }_r={\kappa }_t$ and ${\omega }_i={\omega }_r={\omega }_t$.

[Geometry, part (b)] By step 4.1, ${\kappa }_j=\kappa :={\kappa }_i$ for every $j$, and $k_j={\kappa }_j+(k_j\cdot n)n=\kappa +(k_j\cdot n)n$ by the orthogonal decomposition of [F3]; hence each $k_j$ lies in the complex span of $\kappa$ and $n$, which is two-dimensional over $\mathbb{C}$ when $\kappa \mathrm{\ne }0$ because $\kappa$ and $n$ are independent: if $\kappa =zn$ for some $z\in \mathbb{C}$ then $0=\kappa \cdot n=z(n\cdot n)=z$, so $\kappa =0$, contrary to hypothesis. When the wavevectors are real, $\kappa$ and $n$ are real and each $k_j$ lies in their real span, the corresponding real plane. If instead $\kappa =0$, then every $k_j=(k_j\cdot n)n$ is parallel to $n$.

[Hypotheses used and the absent-wave case, part (d)] The argument of steps 1.1-4.1 used exactly: the two tangential identities of [A1], the nonzero amplitudes $e_j\mathrm{\ne }0$ and the amplitude conditions with the dispersion relation $\epsilon \mu {\omega }_j^2=k_j\cdot k_j\mathrm{\ne }0$ of [F1] (the dispersion enters through the evaluation of $k_j\times h_j$ from $h_j=k_j\times e_j\mathrm{/}({\mu }_j{\omega }_j)$, so the role listed as $k_j\cdot k_j\mathrm{\ne }0$ is the nondegeneracy actually used); the normal orientation conditions $k_i\cdot n>0$, $k_r\cdot n<0$, the two normal interface identities, and the optional normal-root selection rule were not used, and no outgoing or tangential-growth condition was used. If a consuming construction declares the reflected wave absent, $e_r=h_r=0$ (so that $A_r=0$ and only the incident and transmitted traces are matched), the same grouping argument applies to the two nonzero trace data $A_i$ and $-A_t$: a singleton group would give $\pm A_i=0$ or $\mp A_t=0$, impossible by step 1.1, so ${\lambda }_i={\lambda }_t$, i.e. ${\kappa }_i={\kappa }_t$ and ${\omega }_i={\omega }_t$; no constraint on the labels $k_r,{\omega }_r$ of the absent field is produced, and a Fresnel construction may assign it the common phase as a convention.

[Angles and laws, part (c)] Assume now that $k_i,k_r,k_t$ are real. For each $j$, [F1] gives $k_j\cdot k_j=\mathrm{\mid }{k}_j{\mathrm{\mid }}^2={\epsilon }_j{\mu }_j{\omega }^2$ with the common $\omega ={\omega }_j$ of step 4.1, so $\mathrm{\mid }{k}_j\mathrm{\mid }=\omega ({\epsilon }_j{\mu }_j{)}^{1\mathrm{/}2}=\omega \mathrm{/}{v}_j>0$ by [F6] and [F1], and $\mathrm{\mid }{k}_i\mathrm{\mid }=\mathrm{\mid }{k}_r\mathrm{\mid }$. By [F3], $\mathrm{\mid }{k}_j{\mathrm{\mid }}^2=\mathrm{\mid }{\kappa }_j{\mathrm{\mid }}^2+(k_j\cdot n{)}^2$, so $\mathrm{\mid }{\kappa }_j\mathrm{\mid }\le \mathrm{\mid }{k}_j\mathrm{\mid }$ and the ratio $\sin {\theta }_j:=\mathrm{\mid }{\kappa }_j\mathrm{\mid }\mathrm{/}\mathrm{\mid }{k}_j\mathrm{\mid }$ lies in $[0,1]$; define ${\theta }_j:=\arcsin (\mathrm{\mid }{\kappa }_j\mathrm{\mid }\mathrm{/}\mathrm{\mid }{k}_j\mathrm{\mid })\in [0,\pi \mathrm{/}2]$ by [F5], so that $\sin {\theta }_j=\mathrm{\mid }{\kappa }_j\mathrm{\mid }\mathrm{/}\mathrm{\mid }{k}_j\mathrm{\mid }$ as in the Statement. Since ${\kappa }_r={\kappa }_i={\kappa }_t=\kappa$ by step 4.1, $\sin {\theta }_r=\mathrm{\mid }\kappa \mathrm{\mid }\mathrm{/}\mathrm{\mid }{k}_r\mathrm{\mid }=\mathrm{\mid }\kappa \mathrm{\mid }\mathrm{/}\mathrm{\mid }{k}_i\mathrm{\mid }=\sin {\theta }_i$; the principal inverse sine is strictly increasing on its interval because sine is strictly increasing on $[-\pi \mathrm{/}2,\pi \mathrm{/}2]$ by [F5], hence injective, so ${\theta }_r={\theta }_i$. Similarly $\mathrm{\mid }\kappa \mathrm{\mid }=\omega ({\epsilon }_1{\mu }_1{)}^{1\mathrm{/}2}\sin {\theta }_i=\omega ({\epsilon }_2{\mu }_2{)}^{1\mathrm{/}2}\sin {\theta }_t$, that is $n_1\sin {\theta }_i=n_2\sin {\theta }_t$ after multiplying by $c\mathrm{/}\omega$ where $n_j=c({\epsilon }_j{\mu }_j{)}^{1\mathrm{/}2}=c\mathrm{/}{v}_j$ by [F1] and [F6]. If ${\theta }_i>0$ then $\sin {\theta }_i>0$ and division is legitimate, giving $\sin {\theta }_t\mathrm{/}\sin {\theta }_i=({\epsilon }_1{\mu }_1{)}^{1\mathrm{/}2}\mathrm{/}({\epsilon }_2{\mu }_2{)}^{1\mathrm{/}2}=v_2\mathrm{/}{v}_1=n_1\mathrm{/}{n}_2$; if ${\theta }_i=0$ then $\mathrm{\mid }\kappa \mathrm{\mid }=0$, so $\mathrm{\mid }{\kappa }_t\mathrm{\mid }=0$ and ${\theta }_t=\arcsin 0=0$, since $\sin 0=0$ and the principal inverse sine returns the unique number of $[-\pi \mathrm{/}2,\pi \mathrm{/}2]$ with sine $0$, without forming $0\mathrm{/}0$.

[Scope, part (e)] The four statements (a)-(d) are conditional on the given wave data and the interface hypothesis [A1]; nothing about the existence of interface-consistent data, reflection or transmission amplitudes, energy fluxes or well-posedness is proved. All frequencies are positive, all tangential quantities may be complex, and clauses (a) and (b) cover complex transmitted wavevectors; the propagating angle selection of (c) and the amplitudes are the business of the consuming interface theorems. The interface identities on the noncompact plane are the stated hypothesis; for compact interfaces the distributional jump theorem Stationary macroscopic interface jump conditions for D, B, E and H supplies them, and no distributional or choice-theoretic clause of that theorem is used here (only complex arithmetic, the character lemma, and the real geometry of the propagating case), so no choice principle is inherited.