Nonlinear dynamics induced by external optical injection in ...

Nonlinear dynamics in semiconductor lasers 779
then the coupled equations take on a particularly simple form:
dE 1
= 1 2
dt
γ c(E 1 + bE 2 )ñ + A i cos(t) + F 1 (9)
dE 2
=− 1 2
dt
γ c(bE 1 − E 2 )ñ − A i sin(t) + F 2 (10)
dñ
= γ s ( −ñ) − 2ɛ 0n 2 (
g N E
2
dt
¯hω 1 + E 2 )
2 . (11)
0
Here, E 1 and E 2 and F 1 and F 2 are the quadrature field and field noise components,
respectively. The experimentally accessible control parameter is the injection current,
which is normalized for numerical calculation in the parameter = (J − J 0 )/ed/γ s N 0 .
A convenient normalization for the equations can be made by defining R sp = γ c F0
2 and
normalizing the field equations with respect to F 0 . Values for the parameters are derived
from previously reported data [16] with b = 5.5,γ c =5.5×10 11 s −1 ,γ s =9×10 −3 γ c and
2ɛ 0 n 2
g N F0 2 ¯hω = 10−6 γ c . (12)
0
After numerically integrating the normalized, coupled equations, the resulting time series
are Fourier transformed to determine the spectra. Calculated optical spectra are plotted in
figure 9. Using the simplified equations, we can recover the transition from Lorentzian to
near Gaussian and back to Lorentzian as , or the injection current, is increased. Due to the
more simplified model, the agreement is relatively less quantitative than was achieved with
the edge-emitting laser described above. However, the key changes in the spectral profiles
are clearly reproduced. Lorentzian wings are calculated for the near-Gaussian lineshape
at offset frequencies beyond 15 GHz. If, however, the linewidth enhancement factor is
reduced to b = 3, or the strength of the noise power, R sp , is reduced by a factor of 2, the
Gaussian lineshape of the intermediate value in figure 9(b) disappears. In either of these
cases the lineshape remains essentially Lorentzian, with the relaxation resonance sidebands
above threshold, throughout. We should point out that a change in the noise power is
also equivalent to a simultaneous scaling of γ s and g N because of the intimate relationship
between R sp and γ c [27].
Unlike the optical spectrum, the amplitude (intensity) spectrum of the laser field and
the spectrum of carrier density fluctuations are not influenced by a change in the linewidth
enhancement factor. Similarly, for the range of values considered here, they show only
minor changes in shape, accompanied by an overall scaling in strength, due to a change
in the amplitude of the noise source. The calculated spectra for the intensity and carrier
density follow the expected profile from a linearized analysis [17], except at low frequencies
due to the incomplete amplitude pinning near threshold. The amplitude of the laser field
and the carrier density are mutually coupled, but are independent of the phase of the field.
While the phase fluctuations do not couple back to these two quantities, they are strongly
influenced by them through the linewidth enhancement factor and the dependence of the
gain on the carrier density. However, the near-Gaussian lineshape is not a result of nonlinear
dynamics. All three equations can be linearized and the same lineshape will appear. In
a semiconductor laser, the Lorentzian lineshape occurs when the relaxation resonances are
well outside of the central linewidth so that the resonant peaks separated by the relaxation
resonance frequency overlap weakly [3, 28]. Near threshold this approximation will fail. In
cavities characterized by a fast photon decay rate, with a correspondingly large field-noise
source term and a large linewidth enhancement factor like the VCSEL studied here, strong
deviations from the Lorentzian lineshape can be expected.