|
We have now all the information
required to derive the OLCR response of a FBG. The positions and physical
distances are those indicated in Fig. 4-5. The spectral response to a
stationary lightwave of a FBG at its entrance is given by the complex spectral
function rfbg(n) that can be calculated by T-Matrix method (chapter 3, §3.1.4). The
electric field Er from the reference arm on the detector is given by
|
(4-34) |
where Es is the source field,
, x and (1-x) are the
intensity transmission coefficients of the coupler and R is the intensity
reflection coefficient of the reference mirror. The test electric field Et
is given by
|
(4-35) |
where the
propagation constant is only developed to the first order (that is : Dn = 0)
and using equations (4-20) and (4-21a), the following equation is
obtained :
|
(4-36) |
As seen in §4.2.4, Dn = (n(n)-ng(n)). The
intensity I(z) measured by the detector is then
|
(4-37) |
The constant intensity factor Idc
is
|
(4-38) |
The modified coherence function
and the AC amplitude Iac are found
to be
where hfbg(t) is the FBG
impulse response in reflection (Fourier transform of the complex reflection
amplitude rfbg(n)).
The OLCR signal is defined as the
interfering part of I(z), that is
.
Considering a 3 dB coupler (a = 0.5), a perfect reflecting
reference mirror (R = 1) and neglecting the phase factor exp(i2dtk0Dn), the equation
(4-2) is obtained. The OLCR signal is related through the grating impulse
response amplitude hfbg(t) to the reflection amplitude rfbg(n) and for this
reason the logarithmic scale representation is defined as
|
(4-40) |
where the factor 20 takes account for the amplitude
signal.
The resolution in the fiber is defined
as half the coherence length in the fiber
|
(4-41) |
For a Gaussian light source centered at
1300nm with a spectral width of 40 nm propagating in a single mode fiber
with n = 1.45, we find a resolution of Lr = 12.8 mm.
|
