The voltage on the load resistance Z PD of the photodetector is u PD ¼ Z PD Á i PDΣ (R).
The PD current i PDΣ (R) is a sum of all photo-currents i PD (R) with the index R. The
photo-current i PD (R) is the result of multiplication of the emission strength E 12L by
the complex-conjugated quantity E
Ã
12L . Therefore, for the PD current, the expression
is true: i PDΣ R
ð Þ ¼
P
R i PD R
ð Þ ¼ K PD Á
P
R E 12L R
ð Þ Á E
Ã
12L R
ð Þ
Â
Ã
. The PD transfer
function K PD is defined by the expression K PD ¼ |K PD | Á exp (Àj2πfτ PD ), where
τ PD is the time constant of PD.
6.4.1 The Total Transfer Function of the MZ Modulator
We must take into consideration the transfer function of RF electrodes. Expressions
(3.30)À(3.34) for the MZ transfer functions are true when we neglect of losses and
delay of the electrical microwave signal in the MZ system of electrodes. In the case,
when we need to take into account the losses and delay of the electrical signals in the
MZ electrode system, the expression for the MZ transfer function M eZ can be written
as:
M eZ ¼ exp Àα 1m l 1m
ð
Þ
sin α 1m l 1m =2
ð
Þþj sin ξ 1 Á l 1m =2
ð
Þ
½
α 1m l 1m =2
ð
Þþj ξ 1 Á l 1m =2
ð
Þ
½
ð6:7Þ
This expression is calculated with account of propagation of the direct and
reflected electromagnetic waves, which propagates in the strip structure of the
microwave MZ electrodes.
For the module of the MZ transfer function |M eZ |, the following expression is
true:
M eZ
j
j ¼ exp Àα 1m l 1m
ð
Þ
sin
2
α 1m l 1m =2
ð
Þþsin
2 2πf Á n M Á l 1m =2c
ð
Þ
Â
à 1=2
α 1m l 1m =2
ð
Þ
2 þ 2πf Á n M Á l 1m =2c
ð
Þ
2
h
i 1=2
: ð6:8Þ
For the argument of the MZ transfer function (argM eZ ), the following expression
is true:
argM eZ ¼ arctan
sin 2πf Á n M Á l 1m =2c
ð
Þ
sin α 1m l 1m =2
ð
Þ
!
À arctan f Á
2π Á n M =c
ð
Þ
α 1m
!
, ð6:9Þ
where α 1m is the microwave losses in electrodes, l 1m is the electrode length,
ξ 1 ¼ (2πf Á n M /c). Figure 6.11b shows the calculated dependences (6.8), (6.9) of
the module of the MZ transfer function M eZ ¼ M EZ in the small-signal mode. At
that, in Fig. 6.15 the plots correspond to different MZ parameters: (а) α 1m ¼ 0.68
dB/cm, n M ¼ 4.24; (b) α 1m ¼ 0.41, n M ¼ 3.18; (c) α 1m ¼ 0.27 dB/cm, n M ¼ 2.4.
304
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
The PD current i PDΣ (R) is a sum of all photo-currents i PD (R) with the index R. The
photo-current i PD (R) is the result of multiplication of the emission strength E 12L by
the complex-conjugated quantity E
Ã
12L . Therefore, for the PD current, the expression
is true: i PDΣ R
ð Þ ¼
P
R i PD R
ð Þ ¼ K PD Á
P
R E 12L R
ð Þ Á E
Ã
12L R
ð Þ
Â
Ã
. The PD transfer
function K PD is defined by the expression K PD ¼ |K PD | Á exp (Àj2πfτ PD ), where
τ PD is the time constant of PD.
6.4.1 The Total Transfer Function of the MZ Modulator
We must take into consideration the transfer function of RF electrodes. Expressions
(3.30)À(3.34) for the MZ transfer functions are true when we neglect of losses and
delay of the electrical microwave signal in the MZ system of electrodes. In the case,
when we need to take into account the losses and delay of the electrical signals in the
MZ electrode system, the expression for the MZ transfer function M eZ can be written
as:
M eZ ¼ exp Àα 1m l 1m
ð
Þ
sin α 1m l 1m =2
ð
Þþj sin ξ 1 Á l 1m =2
ð
Þ
½
α 1m l 1m =2
ð
Þþj ξ 1 Á l 1m =2
ð
Þ
½
ð6:7Þ
This expression is calculated with account of propagation of the direct and
reflected electromagnetic waves, which propagates in the strip structure of the
microwave MZ electrodes.
For the module of the MZ transfer function |M eZ |, the following expression is
true:
M eZ
j
j ¼ exp Àα 1m l 1m
ð
Þ
sin
2
α 1m l 1m =2
ð
Þþsin
2 2πf Á n M Á l 1m =2c
ð
Þ
Â
à 1=2
α 1m l 1m =2
ð
Þ
2 þ 2πf Á n M Á l 1m =2c
ð
Þ
2
h
i 1=2
: ð6:8Þ
For the argument of the MZ transfer function (argM eZ ), the following expression
is true:
argM eZ ¼ arctan
sin 2πf Á n M Á l 1m =2c
ð
Þ
sin α 1m l 1m =2
ð
Þ
!
À arctan f Á
2π Á n M =c
ð
Þ
α 1m
!
, ð6:9Þ
where α 1m is the microwave losses in electrodes, l 1m is the electrode length,
ξ 1 ¼ (2πf Á n M /c). Figure 6.11b shows the calculated dependences (6.8), (6.9) of
the module of the MZ transfer function M eZ ¼ M EZ in the small-signal mode. At
that, in Fig. 6.15 the plots correspond to different MZ parameters: (а) α 1m ¼ 0.68
dB/cm, n M ¼ 4.24; (b) α 1m ¼ 0.41, n M ¼ 3.18; (c) α 1m ¼ 0.27 dB/cm, n M ¼ 2.4.
304
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .