of QWLD); V a the volume of the active region; e is the electron charge, T OF ¼ τ p is
the photon lifetime in the QWLD resonator.
We note that differential kinetic (balance) QWLD equations (Eq. 7.35) couple by
the cause-and-effect relationship of the inversed population or the carrier density n 1 ,
n 2 , n 3 with the photon flow S p (Eq. 7.35). At deriving, we use the small-signal
approximation for AC components of the pumping currents, of the carrier densities,
of the photon density. Therefore, we use for an analysis at small-signal approximation of the following linearization of the gain in the vicinity of the point with the
values n 1 ¼ n 10 , S p ¼ S 0 :
G n 1 , S p
À
Á ¼ G n 10 , S 0
ð
Þþ
∂G
∂n 1
0 Á Δn 1 þ
∂G
∂S
0
Á ΔS
¼ g 0 þ G n Á Δn 1 þ G S Á ΔS
ð7:36Þ
In Eq. (7.36) G n ¼
∂G
1
ð Þ
∂n 1
0
; G S ¼
∂G
∂S
0
¼ À
G 0 ε sh
1þε sh S 0
; where g 0 is the differential gain
at n 1 ¼ n 10 , S p ¼ S 0 ; ε sh is the gain nonlinearity defined by the spectral “burnout” of
carriers; the symbol “j 0 ” designates that the partial derivative is taken in the point
with the coordinates n 1 ¼ n 10 , S p ¼ S 0 ; in calculations for G
(1) (n 1 ) we use the linear
approximation of the gain maximum: G
(1) (n 1 ) ¼ g 0 (n 1 À n 0 ).
The pumping current I net n 2 , n 1
ð
Þ¼I net n 20 , n 10
ð
Þþ
∂I net
∂n 1 0 Á Δn 1 þ
∂I net
∂n 2
0
Á Δn 2 in
Eq. (7.35), which is determined by the carriers’ capture and rejected from the
quantum well, is defined by the formula: I net n 2 , n 1
ð
Þ¼eV a
n 20
τ c0
À
n 10
τ e0
þ
Δn 2
τ c
À
Δn 1
τ e
,
where τ c0 , τ e0 , τ c , τ e are local time constant of the carrier capture and reject for DC
current and the AC signal.
Now we obtain the QWLD transfer function and PFC and AFC from the velocity
equations (Eq. 7.36). At that, we assume that QWLD operates in the singlefrequency optical generation mode (exactly for this case these equations are true).
In the structure of the small-single direct modulation of the laser emission, the
pumping (bias) current can be presented in the form: I ¼ I 0 + I 1 Á exp ( jωt), where I 0
and I 1 are DC and AC components of the pumping current, and ω is the radio
frequency of the pumping current. Similarly, the photon flow density in the resonator
can be presented as: S ¼ S 0 + S 1 Á exp ( jωt), where S 0 , S 1 are DC and AC
components of the photon flow density in the QWLD resonator. The complex
laser transfer function for the linearized system of the velocity equations is defined
as the ratio: K LD ( jω) ¼ [S 1 ( jω)/S 1 (0)]/[I 1 ( jω)/I 1 (0)].
The expression for the complex transfer function of the mesa-strip quantum-well
laser diode, which operates in the single-frequency mode for the small radio signal of
modulation in its input, can be simplified for modulation frequencies in the range
from 1 to 12 GHz. At that, we can present in the form of the parallel-connected total
capacitance C L and the active differential resistance R d of the complex impedance,
which takes into consideration the constructive parameters of the “laser chip” (the
film thickness, the barrier capacitance, etc.). The differential resistance R d is
7.3 OEO DM Analysis on the Base of Abbreviated Differential Equations
397
the photon lifetime in the QWLD resonator.
We note that differential kinetic (balance) QWLD equations (Eq. 7.35) couple by
the cause-and-effect relationship of the inversed population or the carrier density n 1 ,
n 2 , n 3 with the photon flow S p (Eq. 7.35). At deriving, we use the small-signal
approximation for AC components of the pumping currents, of the carrier densities,
of the photon density. Therefore, we use for an analysis at small-signal approximation of the following linearization of the gain in the vicinity of the point with the
values n 1 ¼ n 10 , S p ¼ S 0 :
G n 1 , S p
À
Á ¼ G n 10 , S 0
ð
Þþ
∂G
∂n 1
0 Á Δn 1 þ
∂G
∂S
0
Á ΔS
¼ g 0 þ G n Á Δn 1 þ G S Á ΔS
ð7:36Þ
In Eq. (7.36) G n ¼
∂G
1
ð Þ
∂n 1
0
; G S ¼
∂G
∂S
0
¼ À
G 0 ε sh
1þε sh S 0
; where g 0 is the differential gain
at n 1 ¼ n 10 , S p ¼ S 0 ; ε sh is the gain nonlinearity defined by the spectral “burnout” of
carriers; the symbol “j 0 ” designates that the partial derivative is taken in the point
with the coordinates n 1 ¼ n 10 , S p ¼ S 0 ; in calculations for G
(1) (n 1 ) we use the linear
approximation of the gain maximum: G
(1) (n 1 ) ¼ g 0 (n 1 À n 0 ).
The pumping current I net n 2 , n 1
ð
Þ¼I net n 20 , n 10
ð
Þþ
∂I net
∂n 1 0 Á Δn 1 þ
∂I net
∂n 2
0
Á Δn 2 in
Eq. (7.35), which is determined by the carriers’ capture and rejected from the
quantum well, is defined by the formula: I net n 2 , n 1
ð
Þ¼eV a
n 20
τ c0
À
n 10
τ e0
þ
Δn 2
τ c
À
Δn 1
τ e
,
where τ c0 , τ e0 , τ c , τ e are local time constant of the carrier capture and reject for DC
current and the AC signal.
Now we obtain the QWLD transfer function and PFC and AFC from the velocity
equations (Eq. 7.36). At that, we assume that QWLD operates in the singlefrequency optical generation mode (exactly for this case these equations are true).
In the structure of the small-single direct modulation of the laser emission, the
pumping (bias) current can be presented in the form: I ¼ I 0 + I 1 Á exp ( jωt), where I 0
and I 1 are DC and AC components of the pumping current, and ω is the radio
frequency of the pumping current. Similarly, the photon flow density in the resonator
can be presented as: S ¼ S 0 + S 1 Á exp ( jωt), where S 0 , S 1 are DC and AC
components of the photon flow density in the QWLD resonator. The complex
laser transfer function for the linearized system of the velocity equations is defined
as the ratio: K LD ( jω) ¼ [S 1 ( jω)/S 1 (0)]/[I 1 ( jω)/I 1 (0)].
The expression for the complex transfer function of the mesa-strip quantum-well
laser diode, which operates in the single-frequency mode for the small radio signal of
modulation in its input, can be simplified for modulation frequencies in the range
from 1 to 12 GHz. At that, we can present in the form of the parallel-connected total
capacitance C L and the active differential resistance R d of the complex impedance,
which takes into consideration the constructive parameters of the “laser chip” (the
film thickness, the barrier capacitance, etc.). The differential resistance R d is
7.3 OEO DM Analysis on the Base of Abbreviated Differential Equations
397