where ω ¼ 2πf, the square of the own RF resonance frequency of QWLD ω
2
L00 and
the QWLD decay decrement μ are functions of the quantum-well parameters. Values
of laser diode capacitance C L , the laser diode resistance R d , the time constant τ ce are
determined by constructive features of the laser diode and are described in Chap. 4.
The fundamental resonance radio frequency is: ω
2
L0 ¼
Г а U gr g 0 P L0
1þR τ r D
ð
Þ1þε sh P L0
ð
Þ τ ph
; and
QWLD decay decrement μ can be written as
μ ¼
1
1 þ R τ r D
ð
Þ τ n1
þ ω
2
L0 τ ph þ
ε sh þ ε ce
ð
Þ1 þ Rr D
ð
Þ
Г а U gr g 0
!
;
ð3:20Þ
where τ ce ¼ r D τ c + τ e ; R τ ¼
τ c
τ e
; r D ¼
L w
L a
; ε ce ¼
Rr D Г а U gr g 0 τ ce
1þR τ r D
ð
Þ
2 ; τ D ¼
L w ÀL a
ð
Þ
2
8D a
; L a is the
total thickness of the quantum well; L w is the thickness of the laser waveguiding
layer; P L0 ¼ S 1L Á (J 0L À J 0Lth ) is the DC component of the photon flow density in
the resonator; Γ a is the coefficient of the optical field limitation; V gr is the group light
speed; D a is the coefficient of ambipolar diffusion; τ c , τ e are the local time of capture
and ejection of carriers for the AC signal, ε sh is the nonlinear gain; g 0 is the gain of
the active layer; τ ph is the photon lifetime in the resonator; S 1L is the proportionality
coefficient defining the dependence of photon flow density (in laser optical resonator) versus the DC pumping current in QWLD; J 0L , J 0Lth are the DC of QWLD
pumping current and its threshold value; τ n1 is the carrier lifetime in the active area.
The module K L and argument φ L of the QWLD transfer function for two different
excesses A 0 ¼ J 0L /J 0Lth of the pumping (bias) current J 0L above the threshold value
J 0Lth are presented in Fig. 3.12. From Fig. 3.12 we see that at growth of DC pumping
current J 0L , the frequency of the resonance peak ω L0 is shifted to the right, and the
peak line width of the transfer function modulus increases, which is related to the
growth of the expression G L ¼ Γ a U gr g 0 P L0 , which is in the numerator, at increase of
the laser DC optical power P L0 . The frequency square is defined by the expression
ω
2
L0 ¼
1
T
2
1
G L Á A 0 À 1
ð
Þ, where T 1 is lifetime of carriers on the upper excited level.
Let us pay attention to the modulation type in DM. In OEO DM, the emission
intensity or the photon flow is modulated. At wave approach, we consider the laser
optical emission as oscillations of normalized EMF strength. Therefore, we speak
not only about the modulation intensity, but also about amplitude modulation of the
strength, i.e., about AM. The mathematical expression of E LL for interfering five
harmonics of the laser EMF on the PD area without application of selection by the
optical filter can be written in the form:
E LL ¼ fE 0L Á I 0 A 0
ð Þcos 2πνt þ ϕ 0O þ φ 0Om t
ð Þ
½
Š
þE 0L Á I 1 A 0
ð Þcos 2πνt þ 2πft þ ϕ 10O þ φ 1Om t
ð Þ
½
Š
þE 0L Á I 1 A 0
ð Þcos 2πνt À 2πft À ϕ 10O þ φ 1Om t
ð Þ
½
Š
þE 0L Á I 2 A 0
ð Þcos 2πνt þ 2 Á 2πft þ 2ϕ 20O þ 2φ 2Om t
ð Þ
½
Š
þE 0L Á I 2 A 0
ð Þcos 2πνt À 2 Á 2πft À 2ϕ 20O À 2φ 2Om t
ð Þ
½
Š g
,
ð3:21Þ
where ϕ 0O , ϕ 10O , ϕ 20O are constant phase shifts defining by the selective optical
filter, φ 0m , φ 10m , φ 20m are phase fluctuations of DC component, first and second
3.2 Methods of Modulation and Heterodyning of Laser Emissions at DM and MZ. . .
99