p
2
þ 1=T eF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
i L ¼ p cos Δϕ OF
½
E n
j j
2 1=T eF
ð
ÞK OF K PD
exp ÀpT FOS
ð
Þ S NY u PD
ð Þ:
ð5:7Þ
We introduce the designation K DL ¼ cos [Δϕ OF ](1/T eF )K OF K PD , and divide the
numerator and denominator of the left part of the equation by (2πf 0e )
2 :
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
Þ p=2π f 0e
ð
Þþ1
h
i
p=2π f 0e
ð
Þ
i L
¼ E n
j j
2 K DL exp Àj2π f 0e T FOS
ð
Þ Á S NY u PD
ð Þ:
ð5:8Þ
We choose 2πf 0e as the reference radian frequency. Replacing p by j2πf 0e + p 1 ,
combining terms in groups according to the smallness order, keeping only the first
order of smallness terms, we obtain the expression for the abbreviated conductance
in the first approximation in the left part of Eq. (5.18) [1]:
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
Þ p=2π f 0e
ð
Þþ1
h
i
p=2π f 0e
ð
Þ
!
j2π f 0e þp 1
1
Q EF
þ p 1
,
ð5:9Þ
where the arrow “ !
j2π f 0e þp 1 ” designates the representation of the controlling conductance in the abbreviated form. At abbreviation, we expand the current in the right
part I 1A into in-phase and quadratic components:
i A u PD
ð Þexp Àj2π f 0e T FOS
ð
Þ !
j2π f 0e þp 1 ½I 1A U PD
ð
Þcos 2π f 0e T FOS
ð
Þ þ
jI 1A U PD
ð
Þsin 2π f 0e T FOS
ð
Þ :
ð5:10Þ
Now we write the abbreviated equation for the current J 10L (where J 10L is the
slowly changing amplitude of the first harmonic of the pumping current oscillations
i L ) of the first approximation, taking into account R L , which is the resistance of the
electrical input of the laser:
1
Q EF
þ 2p 1
exp jΦ 1
½
ÁJ 1L ¼
K 0DL
R L Q EF
I 1A cos 2π f 0e T FOS
½
f
þjI 1A sin 2π f 0e T FOS
½
gexp jΦ 1
½
:
ð5:11Þ
We act in the similar manner as we acted in Sect. 2.4.3 (Chap. 3). For the first
equation of the system (Eq. 5.6), we shall obtain the abbreviated equations for slowly
changing amplitude E 10L and the phase Φ 1L , taking into account that
K 0L Á N 0 (J 1L ) ¼ α 00 Á J 0L + α N01 Á J 1L , R L is the input resistance of the laser, and
5.1 Symbolic and Abbreviated Equations of OEO
211
2
þ 1=T eF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
i L ¼ p cos Δϕ OF
½
E n
j j
2 1=T eF
ð
ÞK OF K PD
exp ÀpT FOS
ð
Þ S NY u PD
ð Þ:
ð5:7Þ
We introduce the designation K DL ¼ cos [Δϕ OF ](1/T eF )K OF K PD , and divide the
numerator and denominator of the left part of the equation by (2πf 0e )
2 :
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
Þ p=2π f 0e
ð
Þþ1
h
i
p=2π f 0e
ð
Þ
i L
¼ E n
j j
2 K DL exp Àj2π f 0e T FOS
ð
Þ Á S NY u PD
ð Þ:
ð5:8Þ
We choose 2πf 0e as the reference radian frequency. Replacing p by j2πf 0e + p 1 ,
combining terms in groups according to the smallness order, keeping only the first
order of smallness terms, we obtain the expression for the abbreviated conductance
in the first approximation in the left part of Eq. (5.18) [1]:
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
Þ p=2π f 0e
ð
Þþ1
h
i
p=2π f 0e
ð
Þ
!
j2π f 0e þp 1
1
Q EF
þ p 1
,
ð5:9Þ
where the arrow “ !
j2π f 0e þp 1 ” designates the representation of the controlling conductance in the abbreviated form. At abbreviation, we expand the current in the right
part I 1A into in-phase and quadratic components:
i A u PD
ð Þexp Àj2π f 0e T FOS
ð
Þ !
j2π f 0e þp 1 ½I 1A U PD
ð
Þcos 2π f 0e T FOS
ð
Þ þ
jI 1A U PD
ð
Þsin 2π f 0e T FOS
ð
Þ :
ð5:10Þ
Now we write the abbreviated equation for the current J 10L (where J 10L is the
slowly changing amplitude of the first harmonic of the pumping current oscillations
i L ) of the first approximation, taking into account R L , which is the resistance of the
electrical input of the laser:
1
Q EF
þ 2p 1
exp jΦ 1
½
ÁJ 1L ¼
K 0DL
R L Q EF
I 1A cos 2π f 0e T FOS
½
f
þjI 1A sin 2π f 0e T FOS
½
gexp jΦ 1
½
:
ð5:11Þ
We act in the similar manner as we acted in Sect. 2.4.3 (Chap. 3). For the first
equation of the system (Eq. 5.6), we shall obtain the abbreviated equations for slowly
changing amplitude E 10L and the phase Φ 1L , taking into account that
K 0L Á N 0 (J 1L ) ¼ α 00 Á J 0L + α N01 Á J 1L , R L is the input resistance of the laser, and
5.1 Symbolic and Abbreviated Equations of OEO
211