Page 46 -

2.2 Theoretical Analysis 35
(a) (b)
Internal cavity External cavity

, b , g, a) (n ,b )
(n 1 1 2 2 (n 1 , b , g, a)
1
r 1 r 2 r ' r ' r 1 r e
2
3
t 2
t ' L
L 2 L ex

Mirror 1 Mirror 2 Mirror 3 z = 0 z = L
Fig. 2.2. Extremely short-external-cavity LD (ESEC LD)system which consists of
the LD facet reflectivities R 1,R 2, and the external mirror reflectivity R 3 (a),and
eff
introduced effective reflectivity R 2 instead of R 2 and R 3 (b). We can analyze the
behavior of the ESEC LD by analyzing the solitary LD with a resonator composed
eff
of R 1 and R 2
R 1 , the LD facet reflectivity R 2 facingthe external mirror, the external mirror
reflectivity R 3 , and the LD drive current. Figure 2.2a shows the ESEC LD
system. The light emits from R 2 , reflects back at R 3 , and returns into the
LD. The returned light changes the carrier density (refractive index) and
the gain spectrum of the internal cavity medium of the LD, which leads to
complicated behavior of the ESEC LD. To analyze this behavior more easily
we introduce the effective reflectivity R eff which has equivalent effect to the
2
coupled R 2 and R 3 . Then we can analyze the behavior of the ESEC LD by
eff
analyzingthe solitary LD composed of a resonator with R 1 and R ,asshown
2
in Fig. 2.2b.
Effective reflectivity R eff has been successfully introduced as expressed in
2
(2.7) to assist the clarification of the lasingcharacteristics of the external-
cavity LD
2 2
2
r + a r +2ar 2 r 3 cos(2β 2 L ex )
∗
3
2
eff eff
eff
R = r r = , (2.7)
2 2 2 2 2 2
1+ a r r +2ar 2 r 3 cos(2β 2 L ex )
2 3
where n 1 is the refractive index of the internal cavity (LD medium), n 2 is that
√ √ √
of the external cavity (air), r 1 = R 1 ,r 2 =
R 2 , r = −r 2 and r 3 = − R 3
2
are the amplitude reflectivities, β i =2πn i /λ is the propagation constant,
√
and a(= η) is the amplitude couplingcoefficient for the external-cavity-
length L ex .
Example 2.2. Derive the effective reflectivity R eff in (2.7).
2
Solution 2.1. The light emits from R 2 , reflects back at R 3 , returns to R 2 ,and
then reflects at R 2 , which leads to multiple reflection in the external cavity.
The returned light amplitude E k after k time reflection is given,
k k−1 k
E k = E 0 t 2 t r r a exp(−j2kβ 2 L ex ), (E2.13)
2 3 2