This post has been overdue for quite a while now. I originally intended to post something about what I do at work soon after I started my new job. That was more than a year ago; time really does fly. Somehow there was always something else already in the pipeline and I never got around to do a proper post on laser Doppler vibrometry (LDV). This is about to change…

The concept of LDV is quite simple: Use an interferometer to determine the phase (and thus the optical path length) of laser light reflected from the device under test (DUT), or test object. If the DUT is moving, especially if it is vibrating, this phase will perform a harmonic oscillation or a superposition of such oscillations, depending on the particular movement of the DUT. Getting this all to work with the best possible performance is not so simple, though, and keeps lots of people at Polytec very busy. However, the exact implementation of LDV is not the topic of this post (and is a trade secret), and we’ll stick to the fundamental principles.

#### Interferometry

Figure 1 shows the principle of LDV using a Mach-Zehnder interferometer (MZI) as an example. We are not limited to this arrangement, though, and a number of interferometric setups can be used for LDV, as will be shown below.

With an MZI, the laser beam is split into two paths by a polarizing beam splitter (PBS). The object, or DUT, is inserted in one of the paths, the other comprises a reference path. In a heterodyne vibrometer, a frequency shifter is inserted into one of the paths – it doesn’t matter which one. In a homodyne vibrometer, this frequency shifter is not needed. There are certain advantages and disadvantages to both methods, some of which are discussed a bit later. The reference and object beams are recombined in an optical hybrid (also called beam splitter or coupler) and the superposed beams are detected by one or two photodetectors. The polarizing beam splitters and the λ/4 plate ensure that the superposed beams are in the same polarization state (otherwise they would not mix coherently in the photodiodes).

The laser beams in the interferometer are spatially confined, usually to a Gaussian intensity profile that is cut off at some radius by an aperture in the laser. In order for the beams in both paths to properly interfere, they must be superposed on the front facet of the photodiode(s). If this superposition is offset or tilted, or if the beams acquire different diameters, the signal strength of the mixing term will deteriorate. However, for the purpose of signal processing, the beam shape is irrelevant and we shall work with simple plane waves. We can just introduce some parameter $0 \le \rho \le 1$ that shall mathematically represent the efficiency of beam superposition. Further disregarding the state of polarization, the optical field of a plane wave with frequency $\omega_0 = k_0 \cdot c$ and free-space wavenumber $k_0 = 2\pi/\lambda$ after propagation of an optical path length$^1$ $z$ is

$$\begin{aligned}

E(z,t) &= \frac{1}{2} \Bigl( A(t) \exp\bigl(i \omega_0 t - i k_0 z \bigr) + \mathrm{c.c.} \Bigr)\\

&= \Re \Bigl\lbrace A(t) \exp\bigl(i \omega_0 t - i k_0 z\bigr) \Bigr\rbrace\\

&= \Bigl|A(t)\Bigr| \cos \Bigl( \omega_0 t - k_0 z + \mathrm{arg}\bigl[A(t)\bigr]\Bigr)

\end{aligned}$$

where $A$ is the complex envelope whose magnitude and phase describe the field at $z=0$ and c.c. represents the complex conjugate of all the terms preceding it (required to make $E$ a real-valued quantity). The field resulting from the reference beam just in front of the optical hybrid can be written as

$$E_\mathrm{ref}(z_\mathrm{BS},t) = \frac{1}{2} \Bigl( A_\mathrm{ref}(t) \exp \Bigl[i \omega_0 t - i k_0 z_\mathrm{BS} \Bigr] + \mathrm{c.c.} \Bigr) \tag{1a}$$

with

$$A_\mathrm{ref}(t) = A_0 \exp \Bigl[i \Delta\omega t\Bigr] \tag{1b}$$

in which $z_\mathrm{BS}$ is the total optical path length up to the hybrid. The complex envelope $A_\mathrm{ref}$ is some constant amplitude modulated by the heterodyne frequency shift $\Delta\omega = 2\pi \Delta f$.

Similarly, the corresponding term for the measurement path is

$$E_\mathrm{meas}(z_\mathrm{BS},t) = \frac{1}{2} \Bigl( A_\mathrm{meas}(t) \exp \bigl[i \omega_0 t - i k_0 z_\mathrm{BS} \bigr] + \mathrm{c.c.} \Bigr) \tag{2a}$$

with

$$A_\mathrm{meas}(t) = A_0 r(t) \exp \Bigl[- i 2 k_0 \Delta z(t)\Bigr] \tag{2b}$$

Its complex envelope is not frequency-modulated, but incurs a phase shift due to the additional optical path $\Delta z$ (compared to the reference path) to the DUT, which varies with its motion. In the setup of Fig. 1, this additional path is traversed twice, hence a factor of 2. Since the moving object usually does not reflect all of the incoming light back into the vibrometer and possibly also causes a modulation of the reflected beam intensity, a real-valued reflectivity coefficient $r$ was introduced. Also, the laser output is assumed to be equally distributed between reference and measurement beam, as reflected by the coefficient $A_0$ in both (1b) and (2b).

The only (intended) change of the measurement signal is a variation of the phase of $A_\mathrm{meas}$ related to the change of the optical path length to the DUT. Hence, the vibrometer will only be able to detect motion along the laser beam. Often this beam is orthogonal to the surface plane of the DUT (this maximizes $r$ especially for mirror-like surfaces) and the so-captured object motion is called *out-of-plane motion*.

#### Detection and Demodulation

The principle of heterodyne detection is explained in detail in the post homodyne versus heterodyne. Following the analysis outlined therein, we will simply insert our fields (1) and (2) to obtain for the difference in photodetector currents, using (6) from that post,

$$\begin{aligned}

I_{1-2}(t) &= R \cdot \Re \Bigl\lbrace \rho A_\mathrm{ref}(t) A^*_\mathrm{meas}(t) \Bigr\rbrace\\

&= \rho \, R \cdot \Re \Bigl\lbrace A_0^* A_0 \, r(t) \exp \Bigl[ i \Delta\omega t + i 2 k_0 \Delta z(t) \Bigr] \Bigr\rbrace \\

&= \rho \, R \, A_0^* A_0 \, r(t) \cos \Bigl[ \Delta\omega t + 2 k_0 \Delta z(t) \Bigr]

\end{aligned}\tag{3}$$

where $\rho$ is the beam overlap efficiency as described above and $R$ is the conversion efficiency of the photodiodes. We can then downconvert the intermediate frequency signal into the baseband and obtain, according to (7) in the other post,

$$B(t) \propto r(t) \exp \Bigl[i 2 k_0 \Delta z(t) \Bigl] \tag{4}$$

Since we don’t need to know the absolute amplitude or some constant phase offset in the signal to determine the DUT motion (the information we need is in the phase modulation), we dropped a few of the coefficients.

To obtain $\Delta z(t)$ from (4), we can use the inverse tangent,

$$\Delta z(t) = \frac{1}{2 k_0} \mathrm{atan2} \Biggl( \frac{\Im\bigl\lbrace B(t) \bigr\rbrace}{\Re\bigl\lbrace B(t) \bigr\rbrace} \Biggr) \tag{5}$$

where atan2 [Wikipedia] is the variant of the inverse tangent that is not ambiguous.

The velocity of the DUT can be obtained by differentiating (5),

$$\begin{align}

v(t) = \partial_t \Delta z(t) &= \frac{1}{2 k_0} \partial_t \mathrm{atan2} \Biggl( \frac{\Im}{\Re} \Biggr)\\

&= \frac{\Re \cdot \partial_t \Im \, - \Im \cdot \partial_t \Re}{B B^*}\tag{6}

\end{align}$$

in which $\Im \equiv \Im\bigl\lbrace B(t) \bigr\rbrace$ and $\Re \equiv \Re\bigl\lbrace B(t) \bigr\rbrace$. Calculating the velocity does not require trigonometric functions or their inverses, and so can be much faster and/or simpler, especially when doing the demodulation digitally. The motion $\Delta z(t)$ can then be obtained from (6) by integration.

In principle, a single photodetector is sufficient and a balanced detector as shown in Fig. 1, subtracting the signals from two photodiodes, is not strictly required – the modulation term is contained in the output of each of the two detectors. However, the well-balanced detector automatically eliminates the constant term as well as the $r^2(t)$ terms in the detector output and also has a better signal-to-noise ratio.

#### Doppler

From all of the above it might not be immediately clear why we call it *Doppler* vibrometry. The Doppler effect was originally discovered as sounds changing their pitch, or frequency, when their source was moving towards or away from the observer. The same happens for light, and the effect is indeed contained in (2). If we have an out-of-plane DUT motion with constant velocity,

$$\Delta z(t) = z_0 + v \cdot t$$

where we arbitrarily set the initial distance $z_0 = 0$ for convenience, (2b) becomes

$$A_\mathrm{meas}(t) = A_0 r(t) \exp \Bigl[- i \Delta \omega t \Bigr] \tag{7a}$$

with

$$\Delta \omega = - 2 k_0 v \tag{7b}$$

and we can see that this is indeed corresponding to a frequency shift which is proportional to the velocity of the DUT, just as in acoustics.

#### Polarization

When the reference and measurement beams are not in the same state of polarization, the coherent mixing term which is extracted by means of subtraction in (3) becomes smaller. To be exact, it scales with the dot product of the Jones vectors describing the state of polarization. In the extreme that both are orthogonal, the current (3) will be zero. We have assumed that both beams have the same state of polarization and thus neglected polarization effects in the analysis above. However, we must make sure that this is really the case.

The setup in Fig. 1 assumes that the frequency shifter changes an input with p-polarization to an output with s-polarization and vice versa. If a frequency shifter is used that does not affect the state of polarization or in a homodyne system, we could e.g. use a polarizing beam splitter instead of the optical hybrid immediately following the laser source.

#### Homodyning

The homodyne version of an interferometer foregoes the frequency shifter and may thus seem to be significantly simpler to implement. However, as outlined in the homodyne versus heterodyne post, a 90-degree optical hybrid is required for unambiguous detection of the measurement signal phase (which contains the DUT motion term). Such a hybrid, in free-space optics, consists of various beam splitters and a phase shifter, which all have to be very well aligned in order to obtain reasonable values for the overlap efficiency $\rho$. Also, four photodetectors must now be implemented and well balanced. Hence, the initially apparent simpler hardware is actually quite complicated.

Apart from the implementation overhead, homodyne versus heterodyne explains in detail how to extract the baseband signal $B(t)$ from the differential photocurrents (see (12) in that post), which is again much simpler since there is no downmixing required. The so-obtained signal is then basically identical to (4) above.

#### Interferometry cont’d

The principles explained above used a Mach-Zehnder-type interferometer for vibration measurement. However, there are other interferometer types which are more or less suited for LDV. Figure 2 shows a Michelson interferometer-based LDV device.

The Michelson interferometer is quite similar to the Mach-Zehnder setup. The DUT is especially easy to integrate into the setup as this kind of interferometer is based on reflection in both paths. Since that also holds for the reference path, the frequency shifter is traversed twice (unless some sort of ring configuration is used) and thus needs to shift the signal only by $\Delta\omega / 2$ each time. Again, λ plates are used to ensure that the states of polarization are matched on the photodetectors. A homodyne setup with a 90-degree hybrid instead of the frequency shifter is also possible.

Other interferometric setups like Mirau are generally also possible, but may be even more difficult to implement, especially in a heterodyne configuration.

**1** Optical path length

last posts in vibrometry:

What about self mixing interferometry ?

I was trying to limit this post to the basics. To understand self-mixing requires detailed knowledge of the processes within a laser diode, which would have exceeded this premise.