# Basic Principles of Vibrometry

At the heart of every Polytec vibrometer system is the laser-Doppler vibrometer – a precision optical transducer used for determining vibration velocity and displacement at a fixed point. The technology is based on the Doppler-effect; sensing the frequency shift of back scattered light from a moving surface.

## The Doppler Effect

If a wave is reflected by a moving object and detected by a measurement system (as is the case with the LDV), the measured frequency shift of the wave can be described as:

fD = 2· v/λ

where v is the object's velocity and λ is the wavelength of the emitted wave. To be able to determine the velocity of an object, the (Doppler-)frequency shift has to be measured at a known wavelength. This is done in the LDV by using a laser interferometer.

## Interferometry

The Laser-Doppler vibrometer works on the basis of optical interference, requiring two coherent light beams, with their respective light intensities I1 and I2, to overlap. The resulting intensity is not just the sum of the single intensities, but is modulated according to the formula

Itot = I1 + I2 + 2 √(I1 I2 cos [2π(r1 - r2)/λ]

with a so-called “interference” term. This interference term relates to the path length difference between both beams. If this difference is an integer multiple of the laser wavelength, the overall intensity is four times a single intensity. Correspondingly, the overall intensity is zero if the two beams have a path length difference of half of one wavelength.

## Experimental Setup

The image above shows how this physical law is exploited technically in the LDV.

The beam of a helium neon laser is split by a beamsplitter (BS 1) into a reference beam and a measurement beam. After passing through a second beamsplitter (BS 2), the measurement beam is focused onto the object under investigation, which reflects it. This reflected beam is now deflected downwards by BS 2 (see figure), is then merged with the reference beam by the third beam splitter (BS 3) and is then directed onto the detector.

As the path length of the reference beam is constant over time (with the exception of negligible thermal effects on the interferometer) (r2 = const.), a movement of the object under investigation (r1 = r(t)) generates a dark and bright (fringe) pattern typical of interferometry on the detector. One complete darkbright cycle on the detector corresponds to an object displacement of exactly half of the wavelength of the light used. In the case of the helium neon laser used almost exclusively for vibrometers, this corresponds to a displacement of 316 nm!

Changing the optical path length per unit of time manifests itself as the Doppler frequency shift of the measurement beam. This means that the modulation frequency of the interferometer pattern determined is directly proportional to the velocity of the object. As object movement away from the interferometer generates the same interference pattern (and frequency shift) as object movement towards the interferometer, this setup cannot determine the direction the object is moving in. For this purpose, an acousto-optic modulator (Bragg cell) is placed in the reference beam, which shifts the light frequency by 40 MHz (by comparison, the frequency of the laser light is 4.74 · 1014 Hz). This generates a modulation frequency of the fringe pattern of 40 MHz when the object is at rest. If the object then moves towards the interferometer, this modulation frequency is reduced and if it moves away from the vibrometer, the detector receives a frequency higher than 40 MHz. This means that it is now possible not only to detect the amplitude of movement but also to clearly define the direction of movement.

## Displacement or Velocity?

In principle the LDV can directly measure displacement as well as velocity. In this case, the Doppler frequency is not transformed into a voltage proportional to velocity; instead the LDV counts the bright-dark fringes on the detector. Using suitable interpolation techniques, Polytec's vibrometers attain a resolution of 2 nm, and with digital demodulation techniques even down to the pm range! Displacement demodulation is better suited for low frequency measurements and velocity demodulation is better for higher frequencies, because the maximum amplitudes of harmonic vibrations can be expressed as follows:

v = 2π • f • s

As its frequency increases, a certain vibration generates higher velocities at lower displacement amplitudes.