= Mode locking
{wiki=Mode-locking}
<Mode locking> fixes relative <phases> between longitudinal <normal modes> of a <laser>, allowing short <optical pulses>. The same <optical spectrum> with wandering relative <phases> would not supply that stable pulse train.<Passion for Precision by Theodor W. Hänsch>[Passion for Precision, p.157: coupled modes]{ref}
Here is an original finite-mode model, with the convention that a positive <frequency> has time dependence $e^{2\pi i f t}$:
$$
E(t)=\sum_{n=24}^{40}a_n e^{2\pi i(n f_r+f_0)t},\qquad
a_n=\frac{e^{-(n-32)^2/18}}{\sum_{j=24}^{40}e^{-(j-32)^2/18}}.
$$
These field coefficients follow a <Gaussian function> and have aligned <phases> at $t=0$. Choose $f_0=f_r/4$ and write $T=1/f_r$. Since $e^{2\pi i n}=1$ for every <integer> $n$,
$$
E(t+T)=e^{2\pi i f_0T}E(t)=iE(t),\qquad |E(t+T)|^2=|E(t)|^2.
$$
The pulse <light intensity> repeats every $T$ although the field needs four periods to repeat. At the pulse centers, $E(0)=1$, $E(T)=i$, $E(2T)=-1$ and $E(3T)=-i$. A detector of <light intensity> alone therefore cannot distinguish this offset from zero. This model has far fewer teeth than a practical optical comb; its normalized time and <frequency> scales are chosen to make the carrier oscillations visible.
\Image[/optical-frequency-comb-pulses-and-teeth.png]
{title=Repeated pulse envelopes can conceal a changing carrier phase}
{description=Original finite sum of 17 <normal modes> weighted using a <Gaussian function>, with $f_0/f_r=1/4$. Both panels represent the same complex field: the upper panel shows its real part and magnitude; the lower panel shows its field-amplitude coefficients. This is a dimensionless model, not historical measurement data.}
A short proof of the converse uses the <Fourier series>. If $E(t+T)=e^{i\Delta\phi}E(t)$, then $E(t)e^{-i\Delta\phi t/T}$ is periodic with period $T$. Its <Fourier series> has <frequencies> $n/T$; restoring the removed factor shifts every tooth by $\Delta\phi/(2\pi T)$. A finite observation time broadens the spectral peaks; it does not change this ideal infinite-train relation.
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