**5.2 Fabry-Pérot interferometer**

In FSI implementation described, the frequency sweep range measurement subsystem is based on a FP interferometer (Vaughan, 1989). As the sweep range measurement is obtained by multiplying the FP FSR by the number of resonances detected while the laser frequency sweeps, the FP interferometer is a critical component of the sensor.

A FP is a linear resonator which consists of two highly reflecting mirrors forming a standing-wave cavity. In this case, we exploit the fact that the transmission through such a cavity exhibits sharp resonances, making it perfect as a frequency reference cavity.

Two basic types of FP cavity exist: planar and confocal. The confocal is more appropriate for the current application as it has inherent higher finesse, is much less sensitive to mirror misalignment (because it is not critical to maintain mirror parallelism), and for the same FSR the length of the confocal cavity is half the length of the planar cavity. When a confocal FP is illuminated by a monochromatic beam close to the axis, a multiple beam interference pattern is produced near the centre of the interferometer. At precisely the confocal spacing, each mirror images the other back upon itself so that a paraxial ray is reentrant after four traversals of the cavity. This means that the transmitted spectrum is reproduced with every quarter wavelength change in the mirror separation. Consequently, the FSR is given by:

$$FSR = \frac{c}{4 \cdot n \cdot d} \tag{38}$$

where *c* is the speed of light, *n* is the refractive index of the optical medium between the mirrors, and *d* is the distance between the spherical mirrors (whose radius of curvature is equal to the spacing between them).

Fig. 6. Typical FSI range and, consecutively, the frequency sweep range value and its uncertainty.

### **5.3 Data processing**

The data acquisition of the FSI interferometer is based in a simple homodyne detection scheme. During the measurement, while the laser frequency is sweeping, signals from the detectors of the FP and interferometer (one or two depending if it is FSI or Dual FSI) are acquired simultaneously. As the maximum number of fringes for the required range is small, it is conceivable to acquire all the data for subsequent processing. After signal processing, we obtain the number of fringes per resonance *N*/*r*.

As the requirements on the sampling rate of the data acquisition subsystem increase, the system should be able to keep only the initial and final parts of the signal with high resolution and acquire the middle data at a lower sampling rate - system performances depend only on the accurate determination of the initial and final resonances and zero crossing positions (another possibility is to lock the laser frequency to the initial and final resonances, although this solution increases complexity).

To retrieve distance from the acquired signals using Eq. (31), the data processing chain must:


where *c* is the speed of light, *n* is the refractive index of the optical medium between the mirrors, and *d* is the distance between the spherical mirrors (whose radius of curvature is

Fig. 6. Typical FSI range and, consecutively, the frequency sweep range value and its

processing, we obtain the number of fringes per resonance *N*/*r*.

resonances, although this solution increases complexity).

4. calculate the number of fringes per resonance.

The data acquisition of the FSI interferometer is based in a simple homodyne detection scheme. During the measurement, while the laser frequency is sweeping, signals from the detectors of the FP and interferometer (one or two depending if it is FSI or Dual FSI) are acquired simultaneously. As the maximum number of fringes for the required range is small, it is conceivable to acquire all the data for subsequent processing. After signal

As the requirements on the sampling rate of the data acquisition subsystem increase, the system should be able to keep only the initial and final parts of the signal with high resolution and acquire the middle data at a lower sampling rate - system performances depend only on the accurate determination of the initial and final resonances and zero crossing positions (another possibility is to lock the laser frequency to the initial and final

To retrieve distance from the acquired signals using Eq. (31), the data processing chain

2. determine the sweep range by measuring the number of Fabry-Pérot resonances within

3. measure the number of synthetic fringes (both integer and fractional part) within the

1. define the sweep range by choosing the initial and final frequency resonances;

equal to the spacing between them).

uncertainty.

must:

the sweep range;

sweep range;

**5.3 Data processing** 

Data processing can be more efficient if all the available data is used. Instead of measuring only *N* and *r* to obtain *N*/*r*, we can measure the number of fringes *Ni* for each resonance *ri* (where *i* represent the sequence of resonances) and determine *N*/*r* using a linear regression, thus reducing the uncertainty in the measurement, and making the procedure more robust to errors (we are making measurements at several tens of resonances instead of only two, the initial and the final).

The control, acquisition and data processing applications were all custom made, implemented in Labview from National Instruments. The system has the capability to do signal acquisition up to 5 MS/s (16 bits) and the output (for laser control) up to 4 MS/s (16 bits).
