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In the T-matrix method [3-2 to 3-4],
the grating is divided in N sections of width Dj (j = 1, …, N), where the parameters Dnac, Dndc
and L are constant. The grating is then defined by N sections with
coupling coefficients qj and physical thickness Dj (Fig. 3-2).
Fig. 3-2 FBG Slicing in sub-sections for the T-matrix method
The knowledge of the fields uj
and vj at the entrance of section j allows to find the fields uj+1
and vj+1 at the layer output. This relation can be expressed in the
form of a transfer matrix relation
|
(3-7) |
where
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(3-8) |
where gj2 = |q j |2 - d2. The fields u1, v1 and uN+1, vN+1
at the grating entrance and output respectively, are then related to each other
by
|
(3-9) |
The reflection coefficient amplitude r(d) is determined
with the limit conditions u1 = 1 and vN+1 = 0 :
r(d) = v1 = -T21/T22.
The transmission coefficient amplitude t(d) is found from the limit
conditions u1 = 0 and vN+1 = 1 :
t(d) = v1 = 1/T22.
The proposed T-matrix formulation takes
into account the overlap integral h that is often neglected [3-1, 3-4]
and the fringe visibility effect can be integrated in the definition of the
refractive index distributions Dndc and Dnac.
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