![](bg9.png)
Figure 16: Effect on axial mode amplitudes with detun-
ing δ f . In the mode amplitude spectra, the mode enve-
lope deviates to the solid line envelope from ideal mode
locked envelope shown in dotted-dashed line. The mode
locations in frequency follow the Fourier frequencies of
( f
n
= n f
0
+ nδ f ), until mode n, that is still in the locking
limit , the further modes (m > n) are unlocked and oscillate
at m f
0
. The locking limit is where the injection signal is in
quadrature to the signal in the cavity as shown in the above
schematic.
the variation in the Re (δA
n
), the amplitude change is
larger the phase change with increasing mode number
n. The modes occur at f
n
= n f
0
, as shown as dotted
delta functions at the mode locations. For detuning
of δ f , the modes occur at f
n
= n f
0
+ nδ f which is a
nδ f frequency deviation from the mode locked state
for each mode. The detuned state is shown with solid
delta functions in frequency spectrum at the bottom
of Fig.16. Once the frequency deviation nδ f is out-
side the locking range R
n
for the mode, the mode is
no longer locked. This is shown for mode m which is
at a higher frequency than the modes n that are within
the locking range. This mode acts as free running os-
cillator, and the frequency of oscillation is same as at
resonance, i.e. n f
0
. However, the phase of the os-
cillation is no longer related to the rest of the modes
((Adler, 1946; Razavi, 2004)). In these modes, the
mode locked behavior is lost as they have frequency
deviation nδ f greater than the locking range R
n
of the
mode. These modes are said to be unlocked.
In the unperturbed, mode locked state, the ampli-
tudes A
n
for all modes are equal. This makes the spec-
tral envelope of the frequency spectrum uniform, as
shown by the dotted horizontal line in Fig.16. On de-
tuning, the spectral envelope deviates from the mode
locked envelope as shown by the solid curve in the
frequency spectrum. The amplitude deviation (re-
ferred to as deviation from here), is a function of n
2
as derived in eq.(12). The δA
n
for each mode is the
change in strength of injection for the mode. When
the mode frequency f
n
is within the locking range R
n
for the mode, the deviation is the difference phasor
between the mode amplitude A
n
at resonance, and the
resulting mode amplitude A
n
+ δA
n
after perturbation
((Razavi, 2004)). The phasors for different modes wrt
the amplitude deviation δA
n
are shown at the top of
the respective mode at n f
0
in the frequency spectrum.
As long as the resulting mode amplitude A
n
+δA
n
and
deviation δA
n
are within 90
◦
of each other, the mode
can be in locked state. The frequency f
n
, in mode is at
the edge of the injection range at either f
(U,n)
or f
(L,n)
.
A condition where the deviation and the resulting am-
plitude are at quadrature to each other is shown for
the mode n in Fig.16. We can see that the maximum
deviation supported by any mode is thus δA
n,Max
, cor-
responding to the limit of locking at either f
(U,n)
or
f
(L,n)
. Mode m greater than n is unlocked, the pha-
sor A
m
+ δA
m
makes an angle greater than 90
◦
with
the deviation δA
m
. The large phase required to main-
tain the angle between the injection and the resultant
mode amplitude cannot be provided by the laser, thus
the mode is not locked and acts as a free running oscil-
lator. The resulting frequency of the unlocked mode
is m f
0
and not m f
0
+ mδ f as would be expected for
a locked mode as is shown in Fig.16. Thus, due to
the large deviation δA
m
> δA
n,Max
, the mode m will
not be locked. In this manner we can relate the injec-
tion range R
n
to a deviation threshold δA
n,Max
for each
mode. In this manner we relate the injection range R
n
to a deviation threshold δA
n,Max
for each mode n.
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