## Basics of Doppler Vibrometry

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.

Fig­ure 1: Laser Doppler vibrom­e­ter in Mach-Zehn­der con­fig­u­ra­tion.

#### 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.

Fig­ure 2: Laser Doppler vibrom­e­ter in Michel­son con­fig­u­ra­tion.

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

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