where l is the waveguide width (Fig. 7.6), ε 20 ¼ ε 2 (y ¼ 0). Modes in the threedimension structure with the two-dimension profile can be divided into modes of two
types, namely, of the E x type (with E y ¼ 0) and the E y type (with E h ¼ 0).
The expression (Eq. 7.10) for nonsymmetrical distribution at (ε 3 À ε 1 ) ¼ 0 transfers into the expression for the symmetrical distribution (Eq. 7.9).
Figure 7.7 presents plots depicted on Eq. (7.10), the permittivity of the optical
waveguide Δε( y) ¼ ε 2 ( y) À ε 20 (0) of the nonsymmetric type for plots is:
1. At (ε 3 À ε 1 ) ¼ 0.101, 2 2ε 20 À ε 1 À ε 3
ð
Þ ¼0:41 (curve 1),
2. At (ε 3 À ε 1 ) ¼ 0.111, 2(2ε 20 À ε 1 À ε 3 ) ¼ 0.382 (curve 2).
If to find the solution in the form:
E x0 h, y, z ¼ x, t
ð
Þ¼E x0 h, y
ð Þexp Àjqx À j2πνt
ð
Þ ,
ð7:11Þ
with the complex propagation constant q, the wave equation for modes E x will have
the form:
∂
2 E h
∂h
2
þ
∂
2 E h
∂y 2 þ
∂
2 E h
∂z 2 þ k
2
0 ε h, y
ð ÞÀq
2
Â
Ã
E h ¼ 0,
ð7:12Þ
where we rename the generally accepted coordinate along the emission propagation
axis as z ¼ x (for utilization of generally accepted designations in the Euler function),
k 0 ¼ 2πν 0 /c.
The equation for modes E y is more complicate, since it contains the term
depending on grad ln ε( y). Therefore, we must be limited in consideration by
modes E x , for which the analytical solution can be presented. We can consider that
in planar strip lasers, configurations of E h and E y along the у axis are identical.
Equation (7.12) is solved by the variables separation. The value q is mainly
determined by the waveguide parameters along the х axis. Then assuming that
E h h, y
ð Þ ¼ E Hh h
ð Þ Á E Yh y
ð Þ,
ð7:13Þ
we obtain [4]:
Fig. 7.7 The permittivity of
the optical waveguide
Δε( y) ¼ ε 2 ( y) À ε 20 (0) of
the nonsymmetric type at
(ε 3 À ε 1 ) ¼ 0.101, 2
(2ε 20 À ε 1 À ε 3 ) ¼ 0.41
(curve 1);
(ε 3 À ε 1 ) ¼ 0.111,2
(2ε 20 À ε 1 À ε 3 ) ¼ 0.382
(curve 2), Functions are
calculated by Eq. (7.10)
7.2 The Model of the Dielectric Waveguide Structure of the Laser and the Optical. . .
379
types, namely, of the E x type (with E y ¼ 0) and the E y type (with E h ¼ 0).
The expression (Eq. 7.10) for nonsymmetrical distribution at (ε 3 À ε 1 ) ¼ 0 transfers into the expression for the symmetrical distribution (Eq. 7.9).
Figure 7.7 presents plots depicted on Eq. (7.10), the permittivity of the optical
waveguide Δε( y) ¼ ε 2 ( y) À ε 20 (0) of the nonsymmetric type for plots is:
1. At (ε 3 À ε 1 ) ¼ 0.101, 2 2ε 20 À ε 1 À ε 3
ð
Þ ¼0:41 (curve 1),
2. At (ε 3 À ε 1 ) ¼ 0.111, 2(2ε 20 À ε 1 À ε 3 ) ¼ 0.382 (curve 2).
If to find the solution in the form:
E x0 h, y, z ¼ x, t
ð
Þ¼E x0 h, y
ð Þexp Àjqx À j2πνt
ð
Þ ,
ð7:11Þ
with the complex propagation constant q, the wave equation for modes E x will have
the form:
∂
2 E h
∂h
2
þ
∂
2 E h
∂y 2 þ
∂
2 E h
∂z 2 þ k
2
0 ε h, y
ð ÞÀq
2
Â
Ã
E h ¼ 0,
ð7:12Þ
where we rename the generally accepted coordinate along the emission propagation
axis as z ¼ x (for utilization of generally accepted designations in the Euler function),
k 0 ¼ 2πν 0 /c.
The equation for modes E y is more complicate, since it contains the term
depending on grad ln ε( y). Therefore, we must be limited in consideration by
modes E x , for which the analytical solution can be presented. We can consider that
in planar strip lasers, configurations of E h and E y along the у axis are identical.
Equation (7.12) is solved by the variables separation. The value q is mainly
determined by the waveguide parameters along the х axis. Then assuming that
E h h, y
ð Þ ¼ E Hh h
ð Þ Á E Yh y
ð Þ,
ð7:13Þ
we obtain [4]:
Fig. 7.7 The permittivity of
the optical waveguide
Δε( y) ¼ ε 2 ( y) À ε 20 (0) of
the nonsymmetric type at
(ε 3 À ε 1 ) ¼ 0.101, 2
(2ε 20 À ε 1 À ε 3 ) ¼ 0.41
(curve 1);
(ε 3 À ε 1 ) ¼ 0.111,2
(2ε 20 À ε 1 À ε 3 ) ¼ 0.382
(curve 2), Functions are
calculated by Eq. (7.10)
7.2 The Model of the Dielectric Waveguide Structure of the Laser and the Optical. . .
379