**2. Interferometry**

Measurements made with interferometers are based on the interference pattern formed by two or more electromagnetic fields. In this discussion, we confine ourselves to the interference of two electromagnetic fields in the visible region of the spectrum. A stable fringe pattern results when the two interfering fields satisfy the following conditions:


To achieve these conditions, the source may be a linearly polarized single mode (temporal and spatial) laser. The light from this laser is split into two beams known as the reference and signal beams. The output optical power resulting from the interference of these two fields takes the form of a raised cosine which varies as a function of the optical path length difference between the reference and signal beams (see Fig. 1).

Fig. 1. Optical output power versus path length difference for an interferometer.

Practical metrology systems require fine resolving capabilities as well as large dynamic range. Fractional fringe interferometers are capable of resolving displacements that are many orders of magnitude less than the optical probe's wavelength. This high resolution is achieved when the interferometer is operated at quadrature, the point of maximum sensitivity. As shown in Fig. 1, operating the interferometer at quadrature yields large changes in the optical output power for small changes in the path length difference between the reference and signal beams. To achieve large dynamic range, interferometers may employ fringe counting techniques for changes in the optical path difference, which exceed half of the field's wavelength.

A number of two beam interferometric configurations have been developed over the past two centuries. In the next section, we will review four of these optical configurations.

#### **2.1 Optical configurations**

318 Modern Metrology Concerns

Measurements made with interferometers are based on the interference pattern formed by two or more electromagnetic fields. In this discussion, we confine ourselves to the interference of two electromagnetic fields in the visible region of the spectrum. A stable fringe pattern results when the two interfering fields satisfy the following conditions:

To achieve these conditions, the source may be a linearly polarized single mode (temporal and spatial) laser. The light from this laser is split into two beams known as the reference and signal beams. The output optical power resulting from the interference of these two fields takes the form of a raised cosine which varies as a function of the optical path length

Section 4 will follow with a summary and directions for future research.

Each of the two fields has its electric field vector linearly polarized.

Fig. 1. Optical output power versus path length difference for an interferometer.

half of the field's wavelength.

Practical metrology systems require fine resolving capabilities as well as large dynamic range. Fractional fringe interferometers are capable of resolving displacements that are many orders of magnitude less than the optical probe's wavelength. This high resolution is achieved when the interferometer is operated at quadrature, the point of maximum sensitivity. As shown in Fig. 1, operating the interferometer at quadrature yields large changes in the optical output power for small changes in the path length difference between the reference and signal beams. To achieve large dynamic range, interferometers may employ fringe counting techniques for changes in the optical path difference, which exceed

 The two fields have the same optical wavelength. The two fields maintain a fixed phase relationship.

The two fields have collinear polarization vectors.

difference between the reference and signal beams (see Fig. 1).

**2. Interferometry** 

In 1881, the American physicist, Albert A. Michelson, developed the basic interferometric configuration shown in Fig. 2. Light from the optical source is split into the signal and reference beams with a beamsplitter. The reference beam reflects off a mirror and retraces its path. The signal beam reflects off the specular surface of the object whose displacement is to be measured. The two beams are recombined using the original beamsplitter. A portion of each of the reference and signal fields is incident on the detector. If the optical path length difference between the reference and signal fields is zero (or an integral multiple of wavelengths), the interference is constructive. If the optical path length difference is *π* (or an odd multiple of half wavelengths), the interference is destructive.

Fig. 2. Michelson optical configuration.

The Mach-Zehnder optical configuration shown in Fig. 3 provides flexibility over the Michelson configuration by using a second beamsplitter to combine the reference and signal beams. With the environment of the reference beam remaining constant, changes in the refractive index through which the signal arm passes may be measured. Because of its suitability for a "push-pull" arrangement, this configuration is used in optical modulator technology (Kaplan & Ruschin, 2000) as well as optical sensor technology (Porte et al., 1999).

Fig. 3. Mach-Zehnder optical configuration.

The Sagnac or cyclic optical configuration is shown in Fig 4. This configuration is unique in that the two beams follow the same path around a closed circuit, but in opposite directions. This sensor measures the non-reciprocal phase changes that arise between light propagating in clockwise and counter-clockwise directions. In gyroscope sensing applications, the phase shift gives a measure of the rotation of the loop about its axis (Saida & Hotate, 1999; Barbour & Schmidt, 2001; Tselikov et al., 1998).

Fig. 4. Sagnac optical configuration.

The optical configuration shown in Fig. 5 is attributed to the French physicist, Hippolyte L. Fizeau. In this configuration, interference occurs between light reflected from the target and the partial reflection from the facet of the fiber probe. High stability (i.e., low phase drift) is a key advantage of this configuration since the fiber/target separation distance is typically very small to accommodate sufficient light coupling back into the fiber. Another advantage of this system is the common path travelled by the majority of the reference and signal beams. In the fiber optic embodiment shown in Fig. 5, this means that the single mode optical fiber does not have to be polarization preserving, since both beams will undergo the same polarization evolution as they travel through the fiber. Each of the interferometric configurations described in this section, encodes a displacement, index change, or rotation rate into a phase shift between two interfering light beams. Practical techniques for extracting the desired parameter from the phase shift information form the topics of the next section.

Fig. 5. Fizeau optical configuration.
