10.4 WDM Devices Based on Grating Principles
415
and n eff is the effective index of the core as described in Eq. (10.32). At this wavelength, the peak reflectivity R max for the grating of length L and coupling coefficient
κ is given by
R max = tanh
2
(κL)
(10.48)
The full bandwidth λ over which the maximum reflectivity holds is [18]
λ =
λ
2
Bragg
π n e f f L
(κ L)
2
+ π
2
1/2
(10.49)
An approximation for the full-width half-maximum (FWHM) bandwidth is
λ FW H M = λ Bragg s
δn
2n core
2
+
L
2
1/2
(10.50)
where s ≈ 1 for strong gratings with near 100% reflectivity, and s ≈ 0.5 for weak
gratings.
For a uniform sinusoidal modulation of the index throughout the core, the coupling
coefficient κ is given by
κ =
πδnη
λ Bragg
(10.51)
with η being the fraction of optical power contained in the fiber core. Under the
assumption that the grating is uniform in the core, η can be approximated by
η ≈ 1 − V
−2
(10.52)
where V is the V number of the fiber.
Example 10.12
(a) The table below shows the values of R max as given by Eq. (10.48) for different
values of κL:
κL
R max (%)
1
58
2
93
3
98