= Evanescent field in total internal reflection
{title2=$k_{2z}=i\kappa$}
The <evanescent field in total internal reflection> oscillates and advances in <wave phase> along the boundary while its <wave amplitude> decays away from it. With the coordinates above and time dependence $e^{-i\omega t}$, a <TE polarization> has <complex amplitude>
$$
\mathbf E(x,z)=A e^{iqx-\kappa z}\widehat{\mathbf y},\qquad
\kappa=k_0\sqrt{n_1^2\sin^2\theta-n_2^2},\qquad \theta>\theta_c.
$$
The physical <electric field> is $\operatorname{Re}[\mathbf E(x,z)e^{-i\omega t}]$. The decaying sign excludes a field that grows without bound at $z\to\infty$. <Dielectric-interface boundary conditions> determine $A$; for nongrazing incidence it is generally nonzero.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Total Internal Reflection and Evanescent Waves, equations (154)–(158), decaying sign and parallel energy flow]{ref}
Zero normal <energy flux> follows directly from <Maxwell equations>. The <wavevector> is $(q,0,i\kappa)$; the relation $\mathbf H=(\mathbf k\times\mathbf E)/(\omega\mu_0)$ gives
$$
H_x=-\frac{i\kappa}{\omega\mu_0}E_y,\qquad H_z=\frac{q}{\omega\mu_0}E_y.
$$
Consequently, the cycle-averaged <Poynting vector> is
$$
\langle\mathbf S\rangle=\frac12\operatorname{Re}(\mathbf E\times\mathbf H^*)
=\frac{q|A|^2}{2\omega\mu_0}e^{-2\kappa z}\widehat{\mathbf x},\qquad \langle S_z\rangle=0.
$$
The <electric field> and $H_x$ are a quarter-cycle apart, so their product has zero cycle average. Tangential <energy flux> survives. This is a local calculation for the single decaying <evanescent wave> in the infinite second <half-space>.
To check reflected <power> independently, put $a=n_1k_0\cos\theta>0$. The existing <transverse-electric Fresnel reflection coefficient> becomes $r=(a-i\kappa)/(a+i\kappa)$. Its numerator and denominator are <complex conjugates>, so $|r|^2=1$, while $1+r=2a/(a+i\kappa)\ne0$ supplies the surface <electric field> relative to the incident <wave amplitude>. Thus a unit <optical reflectance> does not require a zero transmitted field. The same normal <power> balance holds for <TM polarization>; its reflected <wave phase> generally differs from the <TE polarization> phase. A mixture can therefore change its <polarization of an electromagnetic wave> on reflection even though all normal incident <power> returns.
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