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34 2 Extremely Short-External-Cavity Laser Diode
From (E2.1) and (E2.2), ∆λ is given as

2
1 1 1 λ
∆λ =2nL − ∼ = − . (E2.3)
= 2nL −
m +1 m m 2 2nL
The mode interval is the absolute value of (E2.3). The refractive index n is
expressed as
∂n(λ)
n(λ)= n(λ 0 )+ ∆λ. (E2.4)
∂λ
Similarly,
2n(λ 0 )
λ 0 = L (E2.5)
m

2 n(λ 0 )+ ∂n(λ) ∆λ
∂λ
λ 0 +∆λ = L. (E2.6)
m +1
The mode interval ∆λ is given as (E2.6) − (E2.5), namely

1 1 2 ∂λ ∆λ
∂n(λ)
∆λ =2n(λ 0 )L − + L. (E2.7)
m +1 m m +1
From (E2.3)
1 1 λ 0
2
2n(λ 0 )L − = − . (E2.8)
m +1 m 2n(λ 0 )L
From (E2.2),
2L λ 0 +∆λ λ 0
∼
= = . (E2.9)
m +1 n(λ 0 ) n(λ 0 )
Substitute (E2.8) and (E2.9) to (E2.7),
λ 2 0
∆λ = , (E2.10)
∂n(λ)
λ 0
2n(λ 0 ) 1 − L
n(λ 0 ) ∂λ
as n eff is,

λ 0 ∂n(λ)
n eff = n(λ 0 ) 1 − . (E2.11)
n(λ 0 ) ∂λ
Finally, the mode interval ∆λ canbeexpressedas
λ 2 0
∆λ = − . (E2.12)
2n eff L

2.2.2 Effective Reflectivity

The dependence of the lasingcharacteristics of an ESEC LD on external-
cavity length is described using various parameters: the LD facet reflectivity