Remote Laser Interferometry MicrophoneSeth Blumberg, Joel Thompson, David ZaslavskyMay 16, 2006AbstractThis docu ment details the theory and construction of a device that uses laser interferometry to measurethe vibration of a reflective surface (which could be located outside the device itself). Although we havedemonstrated that such a device is buildable and can work in the lab, there are still several issues to be dealtwith before it could become usable in the field for actual measurement.1 Operational Theory1.1 The St ationary Michelson InterferometerThe laser interferometry microphone is based on the Michelson interferometer, shown below:sourcetargetdsdtdmddFigure 1: Schematic of the Michelson interferometer. The laser emitter, beam splitter, reflecting mirror, target,and detecto r are all shown, along with the appr oximate path of the laser beam.The interferometer co mpares the relative phase of two paths of the las er light: (1) the path pr oceeding directlythrough the beam splitter, reflecting off the top mirror , and reflecting off the beam splitter to the detector, and(2) the path initially reflecting off the beam splitter, then r e flec ting off the target, and passing through the beamsplitter to hit the detector. The two paths cover different distances:d1= ds+ 2dm+ ddfor path 1 (1)d2= ds+ 2dt+ ddfor path 2 (2)Therefore, the e le c tric fields of the two beams will probably be out of phase upon arriving at the detector. Usingthe sinusoidal electromagnetic wave equation:~E =~E0sin(kx − ωt + φ) (3)we find that for these two paths, the electric field strengths at the detector are~E1=~E0sin(kd1− ωt + φ) (4)~E2=~E0sin(kd2− ωt + φ) (5)And the overall total electric field is~Ed=~E0sin(kd1− ωt + φ) + sin(kd2− ωt + φ)(6)~Ed=~E0sin(kds+ 2kdm+ kdd− ωt + φ) + sin(kds+ 2kdt+ kdd− ωt + φ)(7)11.2 Changing Path Lengths in the InterferometerThe principle that allows the interferometry microphone to detect sound is the vibration of the target caused byambient s ound waves. This means that the path length component dtvaries in time. For later convenience, weallow dmand~Ed(t) to be time-dependent as well. We can now split the distances dm(t) and dt(t) into constantand variable components:1dm(t) = dm0+ δm(t) dt(t) = dt0+ δt(t)so we now have~Ed=~E0sin(kds+ 2kdm0+ kdd− ωt + φ + 2kδm(t)) + sin(kds+ 2kdt0+ kdd− ωt + φ + 2kδt(t))(8)Equation (8 ) includes terms representing oscillations on two different order s of magnitude. The variationsin the target and mirror dista nce s δt(t) and δm(t) occur with frequencies characteristic of s onic vibrations,approximately 3-5kHz. In contrast, the oscillation of the laser, represented by ω, will be at least several GHz fora maser and nearly 1015Hz for the visible-light lasers that will be available in practice. There fo re we can treatδm(t) and δt(t) as approximately constant over a few oscillations of the laser radiation. Accordingly, we defineour zero point of time and the quantities dm0and dt0such thatkds+ 2kdm0+ kdd− ωt + φ and kds+ 2kdt0+ kdd− ωt + φare integral multiples of 2π. This reduces equation (8) to~Ed(t) =~E0sin(2kδm(t)) + sin(2kδt(t))(9)1.3 Optimizing Intensity VariationOur detector measures the electromagnetic intensity, given byI(t) ∝ E2d(t) (10)I(t) ∝ E20sin(2kδm(t)) + sin(2kδt(t))2(11)Differentia ting I(t), we find that for a small change in δt(t) the intensity variation is∂I∂δt∝ kE20sin(2kδm(t)) + sin(2kδt(t))cos(2kδt(t)) (12)We would like to set up the interferometer to maximize the magnitude of∂I∂δtand thus provide the greatestdetectable signal, so we set∂2I∂δm∂δt∝ 8k2E20cos(2kδm(t)) cos(2kδt(t)) = 0cos(2kδm(t)) = 0 or cos(2kδt(t)) = 0 (13)The left condition leads to the maximum variation, while the r ight condition leads to a minimum of z e ro variation(which would render the apparatus useless). Of cours e, this result is not really practical since we can’t regulatethe position δt(t) to that ac c uracy, but it is useful for the next section.1.4 Feedback Cont rolThis last result presents an immediate problem: if the function cos(2kδt(t)) becomes equal to zero, the intensityI(t) reaches a local extremum and it will be impossible to tell which way it comes back — that is, when thetarget distance changes in such a way as to extremize I(t), there is no way to tell whether the target keepsmoving in the same direction or switches direction, since both cases lead to the same change in intensity.1These δs represent a small changing component of the path length, not the Dirac or Kronecker delta functions, which are notused anywhere in this document.2However, equation (9) sugge sts an alternate method: we can effectively measure δt(t) by altering δm(t) in sucha way as to keep I(t) roughly constant. This requires thatdIdt∝∂I∂δmdδmdt+∂I∂δtdδtdt= 0or4kE20sin(2kδm(t)) + sin(2kδt(t))cos(2kδm(t))dδmdt+ cos(2kδt(t))dδtdt= 0 (14)We can make this zero by maintaining one of these conditions:δm(t) = −δt(t) or δm(t) = δt(t) + π (15)(perhaps in addition to other, unnecessarily complica ted ways). Essentially, the adjustments we make to thepiezo mirror to keep the intensity constant will duplicate (up to a sign) the movements of the target.In practice, we cannot assume the existence of a perfect feedback system, and there is some pos sibility that asudden jump in δt(t) will knock it significantly out of sync with δm(t). However, because the s ine and cosinefunctions are cyclic, we can alter δm(t) in either direction in such a case and it will reach a point that restores theequilibrium before too lo ng. This does present complications with the electronics, though, as described below.2 Construction2.1 The Interferometer SetupThere were a couple of practical issues with our interferometer setup that we didn’t anticipate, which requiredmaking minor changes to the design.sourcepiezo mirrortargetfixedpolarizerspinnablepolarizerFigure 2: Schematic of our actual interferometer setup. This differs from the basic setup (figure (1.1)) only inthe addition of the two polarizers and the contraction of the targ e t distance.The primary problem was that the intensities of the reflected and local beams would be drastically different,since the local be am was bouncing off a full mirror, whereas the reflected beam was bouncing off a partiallyreflective surface (i.e. glass or plastic). This