Compact bulk approximation of the outgoing witness

The witness coefficient $l$ is a phase-adjusted compact-band outgoing quadrature coefficient. Its Fourier packet is not compact in time.

Choose a measurement center $t_{\mathrm{m}}$ and a fixed $T>0$ with

All probe coupling below is thus later than the source switching.

Let

It is an antiunitary involution commuting with real functions of $\omega$. It is not componentwise conjugation of incoming mode coefficients.

For real Cauchy data at $t_{\mathrm{m}}$, define

Put

These are real smooth Sobolev data with compact spectral support away from zero, and $k_{t_{\mathrm{m}}}(q,p)={\omega }^{-2}l$.

Choose $\chi \in {C_c}^{\infty }((-2,2))$, real and equal to one on $[-1,1]$. For $L\ge 1$, set

and let $s_L$ be their free Cauchy solution at $t_{\mathrm{m}}$. Choose a smooth switch ${\eta }_T(t-t_{\mathrm{m}})$, zero before $-T$ and one after $T$, and define

It is compact, with

Write $\delta q=q_L-q,\delta p=p_L-p$, and define