Page 198 - Fiber Bragg Gratings

4.7 Radiation mode couplers 175

the required resonance wavelength. The bandwidth of a single resonance
of an LPG filter is approximately [56]

where coupling from the fundamental guided mode to the appropriate Iv
cladding mode has the core-to-cladding mode effective index difference of
&n eff. The resonance wavelength is determined by Eq. (4.6.4).
Long-period gratings for coupling a guided mode to the cladding may
be designed for standard single-mode fibers by the data in Figs. 4.27 and
4.28. The period of a grating normalized to resonance wavelength A resonance
is shown as a function of the difference of guided mode effective index
from the cladding index. For example, a resonance peak at 1550 nm
requires a grating with a period 100 X 1.55 = 155 microns for a mode-
effective index difference of 0.01.
We note that the change in the resonance wavelength of a long-period
grating is influenced by the change in the core refractive index as a grating
is being written [Eq. (4.7.30)]. As the coupling constant increases, the
coupling loss increases until [Eq. (4.4.12)]

and at the resonance wavelength, a = K ac. To maintain the maximum
loss, the length of the grating or the coupling constant has to be adjusted
for a given bandwidth.
Figs. 4.27 and 4.28 may also be used to show how the period of the
grating varies as a function of the change in the effective index of the
mode. The desired resonance wavelength for an LPG may be calculated
for the final mode effective index after the grating has been fabricated,
using the data in Fig. 4.28 for the real modes of standard SMF fiber.
For a required transmission loss, the refractive index modulation can be
calculated to arrive at the final mode effective index. With this data, and
from the required bandwidth of the grating, the appropriate choice for
the grating period can be made.
Figure 4.29 shows the effect of immersing a long-period grating in oil.
The resonance condition for coupling to the forward-propagating cladding
mode is destroyed as the cladding is index matched. With further increases
in the index of the oil, we note another resonance due to partial reflection