There are some difficulties with reflective fibers displacement sensor which we discuss in the follows . Clearly, the main causes of uncertainty in the fiber displacement measurements performed in vibration testing are:

(i) noise due to the optical source

(ii) shot noise and thermal noise of the receiver

(iii) quantitative errors due to the analog-digital conversion

(iv) computational errors in the numerical deconvolution or in the numerical FFT.

Here we introduced a simple device to decrease the above causes of uncertainty (i)-(iii) terms: The light emitted from the He-Ne laser (with a pulsing driver of power supply, and here we do not use a diode laser) was divided into two parts with different polarization by a cube coupler of the PBS . The set-up is shown in Fig.1 (A), and the light first reflected from the PBS is a reference beam which includes the message of the noise of the optical source. The light transmitted to the PBS is a probe beam, which will be partially reflected when coming into the interface of the bundle of fibers. The reflected probe beam will transmit the PBS for its polarization and neither be reflected to detector 1 to confuse the signal beam nor incident into laser resonator to influence the stability of light source (Fig. 1 (B) ). This set-up can thus avoid the use of a bundle of transmitter and a bundle of receiver fibers and only requires one bundle of fibers. Furthermore, the front slope and back slope problems vanish. The light reflected from the target surface back though the fiber bundle is depolarized as a signal beam proportional to the amount of gap d of displacement between the fiber sensor head and the target surface. The light which reached detector 1 and detector 2 is converted into the electrical signal V_1 and V_2.

Then we obtain:

From this arrangement the noise of the noise terms(i)-(iii) can be depressed to an acceptable range.

The intensity distribution of a laser light spot of TEM_00 mode can be expressed as follows:

where

E(x,y,z) is the light field in the position (x,y,z).

E0 is the intensity factor.

w0 is radius of the waist of a laser beam.

w(z) is radius of the laser spot.

r is the distance from (x,y,z) to the center of the laser spot.

It is reasonable when the laser light is transmitted through the bundle of fibers the Gaussian beam was almost preserved individually but had "broadening effects" together. The "broadening effects" means the peak of the laser light had broadened because of the cross talk terms of every fiber related with each other.

So the intensity of the laser light out of the bundle of fibers can be modified as:

when w < w_b :

when w > w_b :

where

w_f is the radius of the sensor head.

w_b is the diameter of broadening region which depend on the value of d. We can obtain

the value of w_b from the measurement of real intensity profile, but here we only assume the change of the value of w_b is linear as in Fig.2.

The proportion of intensity of the signal received by detector 1 and detector 2 can be calculated by the following integral equation:

where

w(z) = w_0 (d_0+2d)/d_0

d is the distance from the sensor head to the target.

d_0=w_f/tan(theta/ 2) is the distance from the sensor head to the virtue source point.

theta is the divergent angle of the beam.

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