Page 203 - Fiber Bragg Gratings

180 Chapter 4 Theory of Fiber Bragg Gratings

four fields: input fields R(-SL-J2) and R(SL^/2} and output fields S(-SL^I2)
l
and S(8l l/2). A transfer matrix T represents the grating amplitude and
phase response. For a short uniform grating, the two fields on the RHS
of the following equation are transformed by the matrix into the fields
on the LHS as

The boundary conditions applied to Eq. (4.8.1) lead directly to the
reflectivity and transmissivity of the grating. These conditions depend on
whether the grating is a contradirectional or a codirectional coupler.

Reflection grating
For a reflection grating, the input field amplitude R(— $ 1/2) is normalized
to unity, and the reflected field amplitude at the output of the grating
S(^!/2) is zero, since there is no perturbation beyond the end of the
grating.
Writing the matrix elements into Eq. (4.8.1) and applying the bound-
ary conditions leads to

1
in which, the superscript for T has been dropped for clarity. The transmit-
ted amplitude is easily seen to be

The reflected amplitude follows from Eqs. (4.8.2) and (4.8.3) as

Consequently, these are now the new fields on the RHS that can be
2
transformed again by another matrix, T and so on, so that for the entire
grating after the Nih section, where L = E^^-, is