Fundamental principles

A sensor can be defined as a component in which the optical signal will be modulated in response to the measurand. Let's take as an example a source whose spectrum is known, and the electric field  of an optical wave, whose wavelength is . The electric field after the sensor can be written as: :

where  is the transformation matrix of the sensor and X is the vector which defines its environment, including temperature, strain, etc. The configuration of the sensor will enable you to determine T then you just have to invert the previous equation to get the measurand. In a interferometric sensor, the measurand will modulate the phase of the electromagnetic field, which will result in changing the intensity of the interferometer.

We can take T as the product whose effect is observable on the transmitted beam:

where is the scalar transmittance,   is the phase delay and B is the birefringent matrix of the component. ,  et B sont depend on   and on the environment medium. The effects of the matrix B are studied in the next chapter I.2.3.

We can write again the equation (1) with the aid of equation (2) by assuming that the sensor does not modify the polarization of the wave:

The modification of the transmitted wave is obtained with   or .The transmittance   enerally does not depend much on the environment medium, we can assume that is constant, it has been studied in the previous chapter. The sensitivity of the fiber to the three magnitudes of the environment (temperature, strain and pressure) can be written as:

where l is the length of the fiber and n is the effective index of the fundamental mode of the fiber. The first term of the bracket is the physical extension of the fiber and the second is the variation of the effective index.

Most of the interferometers have two waves (i.e. fibers) in which a fiber is subjected to the measurand and another is isolated from it in order to be used as a reference.

For example, let's take an optical fiber strain sensor. To simplify, we will assume that the sensitive element is an isotropic optical fiber with a cylindrical symmetry. We will also assume that the measurand is purely axial without any transverse component. The application of this strain on the fiber will have three effects:

1. the fiber is physically stretched or squeezed.

2. the refractive index of the core and of the jacket are modified, so the effective index of the fundamental mode varies as well.

3. The rays of the core and of the optical jacket will be affected too, consequently the effective index of the fundamental mode will varies as well.

The first effect is the main one, and if we consider the others are negligible, all we have to do is to make the fiber a wavelength longer or shorter to make a one-period change in the interferometer. However, the second effect is about 20% as big as the first one in melted silica and has an opposite sign, so that its sensitivity is a little lower. The third effect is a little bit more complicated. The effective index of the guided modes of the fiber depends on opto-geometrical parameters, like the refractive index of the core and of the optical jacket, on the rays of the core and of the optical jacket and of the considered wavelength. In practice, the fundamental mode's effective index is almost the same as the core's refractive index. By reducing (or increasing) the core's diameter, you can reduce (or increase) the effective index of the fundamental mode, and so come closer (farther) to the refractive index of the optical jacket. However, the third effect is negligible. When you take into consideration all the elements, the sensitivity to strain of a fiber with   is  [21].

A similar study can be carried out concerning the temperature, then the three effects are:

1. 1.fiber lengthening due to the heat

2. modification of the refractive indexes of the fiber because of the thermo-optic effect

3. the rays of the fiber increase due to the heat

In melted silica the thermal expansion coefficient is very low, consequently only the second effect has an impact. When you take into consideration all the elements, the thermal sensitivity of a fiber with   is for a sensitive one-meter element [22].

We can also study the effect of pressure which will reduce the geometrical dimensions (length and diameter) and modify the refractive indexes via the elasto-optic coefficient [23].

We know that interferograms are periodic with a rad period in terms of phase difference or of optical path length of . In order to infer the value of the measurand as precisely as possible, you have to know for sure the optical path length, but this is not easy given the interferogram's periodicity.