Theory of Higher-Order Hermite-Gaussian Pulse Generation From an FM Mode-Locked Laser

We describe theoretically the generation of higher-order Hermite-Gaussian (HG) pulses by using FM mode-locking with a specific optical filter
$F_{HGm}({\omega })$
, which is characterized by
$A_{HGm}({\omega })$
and
$A_{HGm}$
(
${\omega }+{n} {\Omega }_ {m} $
) with
${n}=-{\infty } {\sim } {+\infty }$
. Here,
$A_{HGm}({\omega })$
is the Fourier transformed spectrum of the
$m$
th HG pulse
$a_{HGm}(t)$
in the time domain, and
${\Omega }_ {m} $
is the fixed angular phase modulation frequency. To prepare each filter function
$F_{HGm}({\omega })$
for generating stable HG pulses, we newly developed a T-M map method to make it possible to choose an appropriate combination of
$M_{PM}$
and
$T$
. By selecting their suitable combinations, we successfully generated
$m= 1, 2, 3, 5$
, and 10 HG pulses.
$m= 2, 4$
,
$6 {\ldots }$
, namely even numbered HG pulses, have a center frequency mode whereas 1, 3,
$5 \ldots $
odd numbered HG pulses do not. Finally, we generated dark and bright HG pulses, where we showed that four kinds of HG pulses, which we called positive bright, positive dark, negative bright, and negative dark HG pulses, could be obtained with four combinations of ±HG pulses and ± rectangular pulses. According to T-M map analyses, a condition of
$M_{PM}{\not =}2.4$
was mandatory for these pulses except for bright even HG pulses such as
$a_{HG{0}}{}^{pb}(t)$
,
$a_{HG{2}}{}^{pb}(t)$
,
$a_{HG{4}}{}^{pb}(t), {\ldots }$
since they require a center mode with a large amplitude. This is in contrast to pure HG pulse generation.