Phase modification: how does it works?
The sensitivity of the sensor depends on the rotation degree of the polarization induced by the measurand and on the minimal detectable rotation. The influence of the measurand on the state of polarization can determined as follows: the phase of a wave guided by a fiber whose length is called L is :
where
is the propagation constant of the mode, neffis its effective index and
is the wave vector with
as wavelength. The phase difference between two modes guided by the fiber after a L is :
where
est la différence entre les indices effectifs des deux modes de polarisation.
Effect of mechanical strain
Let's assume that the fiber be subjected to an external mechanical strain called
. The phase difference produced by
is proportional to the fiber's length which is subected to this strain. So it is important to consider the answer of the detector unit of length after unit. Then we have :
where D and n are respectively the transverse dimension and the index profile of the fiber.
The first term of equation (3) describes the photoelastic effect, that is to say the variation of the refractive index of a material according to the mechanical strain. In the case of normal circular fiber, this variation can be calculated with:
where
is the Poisson's ratio of silica which is supposed to be identical for the core and the cladding layer, and p11, p12
are the elasto-optic coefficient of silica [42].
The second term of equation (3) is related to the variation of the fiber's portion when it is subjected to a mechanical strain. This modification of the section will affect the effective indexes of the modes and so their differences as well (i.e.
). It has been proven that it does not affect the phase difference's variation much, so that in practice, it is negligible.
The last term of equation (3) describes the variation of the fiber's length due to the mechanical strain.
Effect of temperature
We can do a study similar to the previous one in order to describe the modifications induced by temperature. The variation of the phase difference due to temperature (T) can be obtained with:
Equation (5) has the same first two terms as equation (3), they are related to the modifications of the fiber's opto-geometrical parameters ( n and D ) which induce a variation of
. The variations of the refractive index are due to the termo-optic effect. The third term represents here the compression of the thermal expansion, which is:
where
is the thermal dilatation coefficient.
The variation of the phase difference is essentially due to the modification of the refractive index according to temperature.
Equations (3) and (5) are used to calculate the mechanical and thermal sensitivity of circular fibers. As for birefringent fibers, calculating it is more complicated since the forms and the material which make up the fiber are more complex. So people often have to experimentally determine the sensitivity of the most exotic fibers.