= Coherent harmonic pulse-train model
{title2=$I(t)\propto[\sin(N\omega t)/\sin(\omega t)]^2$}
Choose $N$ equal-amplitude, monochromatic odd components with orders $q_j=q_0+2j$, $j=0,\ldots,N-1$, and odd $q_0$. Fix the complex-field convention
$$
\mathcal E(t)=a\sum_{j=0}^{N-1}e^{i\phi_j-iq_j\omega t},\qquad I(t)\propto|\mathcal E(t)|^2.
$$
Here $I$ is a normalized <light intensity> after averaging the fast carrier oscillations. Setting every $\phi_j=0$ and summing a <geometric series> gives
$$
\frac{I(t)}{N^2|a|^2}=\left[\frac{\sin(N\omega t)}{N\sin(\omega t)}\right]^2.
$$
At the removable singularities take the continuous limit. Maxima repeat every $T/2=\pi/\omega$; the first zeros on either side of a maximum lie at $t=\pm T/(2N)$. The zero-to-zero width is $T/N$, which is not the <full width at half maximum>.
Choose orders $13,15,17,19,21$ and driving <wavelength> $\lambda=800\,\mathrm{nm}$. Using $T=\lambda/c$ with the <speed of light> $c$ gives $T=2.66851\,\mathrm{fs}$ and burst spacing $1.33426\,\mathrm{fs}$. Solving the exact half-maximum equation gives a <full width at half maximum> of approximately $240.6$ attoseconds. These are values of the specified five-component model, not measurements from the 2001 experiment.
A deliberately chosen <spectral phase> $\phi_q=q\omega t_0$ translates the whole <attosecond pulse train> by $t_0$ without changing its shape. Adding $\kappa(q-17)^2/2$ makes neighboring phase increments unequal: the same component intensities then produce a different temporal profile. For the choice $t_0=0.20\,\mathrm{fs}$ and $\kappa=0.08$ radians per squared harmonic order, the adjacent-component delay estimates are
$$
\frac{\phi_{q+2}-\phi_q}{2\omega}
=t_0+\frac{\kappa(q+1-17)}{\omega},
$$
namely $0.0981$, $0.1660$, $0.2340$ and $0.3019\,\mathrm{fs}$ for $q=13,15,17,19$. This <finite difference> estimates the <group delay> at the pair's midpoint. It is not a single common arrival time.
\Image[/coherent-harmonic-train-and-phase-delays.png]
{title=Identical harmonic intensities can encode different pulse shapes and delays}
{description=Original idealized calculation for five monochromatic odd components. The top panels compare equal component intensities and the resulting normalized <light intensities>. The lower panels compare a chosen delay-dependent two-path signal and the <group delays> recovered from neighboring <spectral phases>. All values are stipulated model inputs or calculated outputs; none are experimental traces.}
The infinite monochromatic sum is periodic forever and has infinite total <energy>. A real <attosecond pulse train> has a finite envelope and finite component linewidths. This calculation isolates coherent addition; it cannot describe an isolated burst or establish its contrast against satellites. As a further check, averaging $I(t)$ over one half-period gives $N|a|^2$ for any constant component <spectral phases>, because the cross terms integrate to zero by <Fourier orthogonality>. <Spectral phase> redistributes the <light intensity> in time while leaving that average fixed.
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