= Two-path attosecond phase reconstruction
{title2=$S_q(\tau)=C_q+D_q\cos(2\omega\tau-\Delta\phi_q+\delta_q)$}
An original two-amplitude model makes the timing inference explicit. Use the <spectral phase> convention of the <coherent harmonic pulse-train model>, and define positive delay $\tau$ to retard the reference <electric field>. Let the amplitudes leading to the same final <photoelectron> state be
$$
M_-(\tau)=b_-e^{i(\phi_{q-1}+\omega\tau+\eta_-)},\qquad
M_+(\tau)=b_+e^{i(\phi_{q+1}-\omega\tau+\eta_+)},
$$
with real $b_\pm\geq0$. Squaring their sum gives
$$
S_q(\tau)=b_-^2+b_+^2+2b_-b_+\cos(2\omega\tau-\Delta\phi_q+\delta_q),
\qquad \Delta\phi_q=\phi_{q+1}-\phi_{q-1},\quad\delta_q=\eta_--\eta_+.
$$
The $2\omega$ oscillation comes from opposite signs of the reference <photon>'s <phase>. For $\Delta\phi_q=2\omega t_0$ and $\delta_q=0$, the signal peaks at $\tau=t_0$ modulo $T/2$. A fitted phase contains the target contribution $\delta_q$; setting it to zero is an assumption, not an observation. In an experiment, unobserved final channels can add as probabilities, so the displayed formula is one effective-channel idealization.
For a chosen example, impose $\Delta\phi_q=0.80$ radians and $\delta_q=0.15$ radians at $800\,\mathrm{nm}$. The observed oscillation has argument $2\omega\tau-0.65$. Its peak shift is $0.1380\,\mathrm{fs}$, while the corrected finite-difference <group delay> is $\Delta\phi_q/(2\omega)=0.1699\,\mathrm{fs}$. Ignoring the target contribution would bias the inferred delay by $-31.9$ attoseconds. These stipulated phases demonstrate a reconstruction error; they are not atomic calculations or measured delays.
Recovering all neighboring phase differences determines the component <spectral phases> up to an overall <phase> and the chosen timing origin, provided the oscillation phases are consistently unwrapped. Known component amplitudes then give a temporal field by a finite inverse <Fourier transform>. This is why measuring the <optical spectrum> alone is insufficient: it omits the very information that decides when constructive <wave interference> occurs.
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