Optical heterodyne accelerometry: passively stabilized, fully balanced velocity interferometer system for any reflector William T. Buttler1,* and Steven K. Lamoreaux2 1
Los Alamos National Laboratory, Physics Division (P-23), MS H803, Los Alamos, New Mexico 87545, USA 2
Yale University, Physics—SPL, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA *Corresponding author:
[email protected] Received 25 March 2010; revised 11 June 2010; accepted 10 July 2010; posted 22 July 2010 (Doc. ID 125765); published 6 August 2010
We formalize the physics of an optical heterodyne accelerometer that allows measurement of low and high velocities from material surfaces under high strain. The proposed apparatus incorporates currently common optical velocimetry techniques used in shock physics, with interferometric techniques developed to self-stabilize and passively balance interferometers in quantum cryptography. The result is a robust telecom-fiber-based velocimetry system insensitive to modal and frequency dispersion that should work well in the presence of decoherent scattering processes, such as from ejecta clouds and shocked surfaces. © 2010 Optical Society of America OCIS codes: 120.4640, 120.7250, 290.5880.
1. Introduction
Shock physics presents special difficulties in measuring the particle velocities of materials experiencing high strains. Early solutions to this problem provided surface velocimetry through the use of poor temporally resolved diagnostics, such as high-speed flash photography and radiography or time of arrival diagnostics, such as shorting pins or piezoelectric transducers. With the advent of the laser in 1958 [1], optical heterodyne velocimetry techniques quickly followed as early practical uses of lasers [2–5]. These early optical/laser Doppler velocimetry (LDV) techniques typically included simple optics, such as balanced 50=50 beam splitters (BSs) and mirrors to form an interferometer and a photomultiplier tube to monitor the interference beat frequency. Within two years of the early Doppler work [2,3], Barker and Hollenbach demonstrated LDV normally reflected from materials under uniaxial strain [6], a 0003-6935/10/234427-07$15.00/0 © 2010 Optical Society of America
demonstration that determined particle velocities of the order of 102 m=s with a Michelson interferometer and fast (for the time) photodetectors. The implication of this result for the advancement of shock physics was revolutionary, but the direct optical Doppler techniques at the time were limited by the available shorter optical laser wavelengths, detector technology bandwidths with their noise levels, and signal recording technologies. Barker cleverly addressed the detection and recording limitations using an unbalanced Mach–Zehnder (uMZ) interferometer similar to that seen in Fig. 1 [7]. The realization of the apparatus formed an optical heterodyne accelerometer that superposes (interferes) early Dopplershifted continuous-wave (CW) laser light with Doppler-shifted CW laser light formed at a later time. This basic concept has been improved considerably over the decades and has come to be the optical diagnostic commonly known as the velocity interferometer system for any reflector (VISAR) [8–11]. With the development of optical communication laser systems, which incorporate high-power longwavelength laser systems (e.g., single frequency and single spatial mode fiber-laser systems with 10 August 2010 / Vol. 49, No. 23 / APPLIED OPTICS
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Fig. 1. This VISAR-like geometry presents an optical heterodyne accelerometer, where early light that travels the long path is superposed with later light that travels the short path. (Subsequent improvements to the early concept, which include an etalon, polarizers, and wave plates, are omitted.)
λ ¼ 1550 nm), high-bandwidth infrared detectors and oscilloscopes (recording systems), direct optical Doppler measurements of high velocities from materials under high strain are now possible [12,13]. However, while these systems have been demonstrated to directly determine velocities up to a few 103 m=s, they have difficulty resolving slower velocities attributable to elastic precursors of shocked materials in the presence of sharply increasing velocities. This is in part due to the data processing techniques, which typically use large Fourier windows to increase the signal-to-noise ratio by averaging within the signals, and in part due to the longer beat times relative to the rise times of the leading shocks, i.e., the velocity rise time is faster than the time of one beat. On the other hand, VISAR systems work well to determine the slower elastic precursor velocities together with the higher jump velocities up to the order of 104 m=s, so long as the surface does not liquefy when the material is shocked (subjected to an extreme, intense pressure wave impulse—shock wave), or liquefy when the shock wave passing through the material releases to zero pressure at the metal– vacuum interface, and so long as the velocity jump does not happen so quickly that the beat frequency of the reference beam with the Doppler shift exceeds the bandwidth of the detection and recording systems [14]. In the situation where the surface of the shocked material is liquid in the release state, and where the scattered signal continues to be collected, the VISAR noise increases dramatically, with the noise increase typically attributed to multiple velocities that causes phase noise within the reflected signals that reduces the spatial and temporal coherence further. In contrast, direct Doppler techniques usually continue to perform in the presence of multiple velocities from liquefied surfaces, so long as any ejected material is not optically thick [15]. In this paper, we present a VISAR-like velocity differencing interferometer that has its length, field, and intensity modes passively stabilized with an 4428
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interferometric concept developed for use in fiberbased quantum cryptography. For example, early fiber-based quantum-cryptography concepts linked two uMZs by single-mode optical fibers [16–18], as illustrated in Fig. 2. One uMZ was used as a transmission state-preparation optic and the other as the reception and measurement optic. These separate and fiber-linked uMZs required active balancing and stabilization because of environmental fluctuations, such as temperature and vibration, which cause polarization and phase drifts that reduce fringe visibility. The elegant solution to these problems was to passively balance the field modes and stabilize the uMZs in a self-correcting, orthoconjugating [19] system that is known as plug-and-play (autocompensating) quantum cryptography [20–22], as shown in Fig. 3. To form a stable velocity differencing interferometer, we incorporate the autocompensating elements of self-stabilization and the passive, near balancing of the VISAR-type uMZ, as shown in Fig. 4, where the optical diagnostics are on the left, and the dynamic (shocked) reflector is to the extreme right. The optical system includes a 1550 nm telecom-fiberlaser that transmits CW horizontally polarized light (h), optical circulator (C), three polarizing beam splitters (PBSs), single BS, 90° half-wave retarder (λ=2), ^ [23], 45° Faraday rotator polarization operator (P) (F) [24], a flyer plate or dynamic reflector (shocked sample), and four detectors (Det) used in pairs for quadrature detection [25] of the interference amplitudes aðtÞ; bðtÞ; a0 ðtÞ; b0 ðtÞ; the system functions similarly to the autocompensating optical system in Fig. 3. That is, the h-polarized CW light is divided into the long and short arms of the uMZ and transmitted as a mixture of h- and v-polarized superposition through the F that rotates the polarizations by 45°. The light then reflects from the shocked sample back along the optical path and is rotated an
Fig. 2. Unbalanced Mach–Zehnder geometry is characterized by a long and short arm for each Mach–Zehnder. The sender prepares and divides a dim, weak coherent pulse in two (“single-photon” superposition), adds a random phase ϕ, and sends the photon to her cohort who adds a random phase β used to determine the random phase ϕ. Simple analysis reveals the “single-photon” can travel either the short-short, short-long or long-short, or the long-long paths, with interference occurring at the middle time only with unbalanced intensities.
ψ 0 ¼ A0
Fig. 3. In the autocompensating geometry, a horizontally (h) polarized “single photon” is transmitted from left to right through the fiber circulator (C) to the BS that causes a superposition. In the upper arm, the λ=2 wave plate rotates the h polarization to vertical (v) before passing the superposition through the PBS toward the optics on the right that reflect the photon before adding a random phase β and directing the photon back to the transmitter for phase analysis. Because the polarizations were rotated by the orthoconjugating Faraday mirror (FM) [24], the paths are passively switched by the PBS, balancing and stabilizing the interferometer.
additional 45° when it again passes through F, so that light that traveled the short path on transmission travels the long path on return, and light that traveled the long path on transmission travels the short path on return, passively balancing and stabilizing the arm lengths of the interferometer and interfering intensities. The functional operations of ^ in the uMZ short arm, the λ=2 in the uMZ upper P arm, and the F with the shocked sample are described in Appendix A. 2. Physics and Optical Model
Consider ψ 0, a horizontally polarized laser beam traveling left to right between the laser and the BS entrance to the uMZ:
1 gðτ; ω0 Þ ¼ A0 ½hgðτ; ω0 Þ; 0
where ω0 is the frequency of the laser light, and τ ¼ t þ z=c is the relative time from the formation of the laser pulse to its detection after traveling over distance z divided by the group velocity of light c, and gðτÞ is the laser-light correlation function, for which the light is assumed to be correlated over times much greater than Δt ¼ 2L=c, the temporal difference between the upper and the lower arms of the uMZ. As ψ 0 (h-polarized light) transmits from left to right, the circulator C transmits h directly through PBS1 toward the BS that divides ψ 0, forming a correlated superposition as half of the h reflects to the longer upper path, and the other half transmits to the shorter lower path of the uMZ. In the upper path, the λ=2 ^ transrotates the h to v, while in the lower path, P mits h unchanged, so that after the beams in the upper and lower paths emerge from the uMZ past PBS2 , ψ 0 ↦ψ 1 þ ψ 2 ¼ ψ: A0 1 A ψ 1 ¼ pffiffiffi gðτ; ω0 Þ ¼ p0ffiffiffi ½hgðτ; ω0 Þ; 2 0 2 0 A A ψ 2 ¼ p0ffiffiffi gðτ þ Δt; ω0 Þ ¼ p0ffiffiffi ½vgðτ þ Δt; ω0 Þ: 2 1 2
ð2Þ
In this arrangement of optical elements, v-polarized ψ 2 lags h-polarized ψ 1 by Δt. At the end of the optical channel is, typically, a lens (collimated or not) that transmits the channel light through F and across the air gap to reflect from the shocked material. If we first consider the case where the reflector is not moving (has not yet been shocked), then the early h portion of ψ reflects from the stationary surface first, while the late v portion of ψ reflects from the surface after time Δt later. After reflection, the light then transmits back through F and is recollected into the optical channel (fiber) and directed back toward the uMZ for analysis. On its return, but prior to entrance into the uMZ at PBS2 , we have light in the polarization state of ψ 0 ¼ ψ 01 þ ψ 02 : A ψ 0 ¼ p0ffiffiffi ð½vgðτ; ω0 Þ þ ½hgðτ þ Δt; ω0 ÞÞ; 2
Fig. 4. These elements define an autocompensating telecomfiber-based optical heterodyne accelerometer; the individual elements are described in detail in the text and legend. Optical paths where light travels in both directions are marked with doubleended arrows, and optical paths where light travels in only one direction are marked with single-ended arrows.
ð1Þ
ð3Þ
where now v-polarized light is leading h-polarized light because the F has rotated the polarizations by a total of 90°. At this point it is clear that ψ 01 will reflect into the upper path, and ψ 02 will transmit into the lower path (see Appendix A), and when the light reaches the BS, the interferometer is passively stabilized to environmental fluctuations that cause the upper and lower path lengths and the refractive indices (birefringence) of the fiber to vary with time, all the polarization and spatial modes overlap, and the intensity is balanced because the beam superpositions have traveled identical paths to and from the reflector. 10 August 2010 / Vol. 49, No. 23 / APPLIED OPTICS
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The situation changes if the reflector is moving, which is the case of interest and the general application. In this case, the optical system will deliver light to a target, or material sample, that will be subjected to a high-pressure impulse or shock wave, as illustrated in Fig. 5. In a typical shock physics experiment, a material of interest might have a highexplosive (HE) charge press fitted in contact with the sample. Once initiated, the HE detonates and drives a shock wave into the material, causing the sample to suddenly move at a high velocity, a velocity of the order of 1 mm=μs. To understand this case, consider the physics described by Fig. 6. In this figure, light is traveling from left to right in the Lagrangian reference frame, meaning the entrance BS is to the extreme left, the shocked sample near the middle of the system, and the BS is again to the extreme right. When the reflector is in motion, the light that enters the uMZ along the short path travels a longer path than the light that first travels the upper path (because the reflector is moving toward the uMZ). Thus, the light that enters the upper path reaches the reflector later, but reaches the interference BS first, defining the time difference between the interfering signals. In matching the path lengths backward from the BS, we find P1 − 2Δx ¼ 2ðx1 þ L − ΔxÞ ¼ P2 , when the sample is in motion. With these physics we can determine the relative phase at the BS: x − 2Δx x þ 2L ϕ1 ¼ ω0 1 þ ω1 1 ; c c x þ 2L − Δx x − Δx þ ω2 1 ; ϕ2 ¼ ω0 1 c c
Δϕ ¼ ðω0 − ω1 Þ
2L Δx x þ ðω0 − ω2 Þ þ ðω2 − ω1 Þ 1 : c c c ð6Þ
Equation (6) can be rewritten as 8 > >
2 1 x1 > : ω0 uc1 Δt − u2 −u c c
where, under the assumption of normal incidence and reflection, the Doppler-shifted frequencies are uðτÞ ; ω1 ≡ ωD ðτÞ ¼ ω0 1 þ 2 c uðτ þ Δt0 Þ 0 ; ω2 ≡ ωD ðτ þ Δt Þ ¼ ω0 1 þ 2 c
Fig. 6. Interference path lengths P1 and P2 demonstrate that light formed at an earlier time interferes with light formed at a later time. The paths define the relevant Doppler-shifted angular frequencies, and the relative path lengths that determine the relative phase difference, relating velocities. (Individual optical elements within P1 and P2 are defined within the text.)
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if terms that scale as δt ¼ Δx=c are ignored [26], demonstrating that Δϕ ¼ const relative to the initial phase ϕ0 when u1 ¼ u2 (ϕ0 is a constant needed for integration). When u1 ≠ u2 , Eq. (7) can be rearranged to show that uðτ þ ΔtÞ − uðτÞ Δϕ T ≈~ aðτÞT ¼ uðτÞ þ K ; Δt 2π
ð8Þ
where ~ aðτÞ is the acceleration of the surface over time Δt, T ¼ x1 =c, and where the velocity per fringe constant K ¼ λ0 =2Δt relates the increase in velocity of the surface relative to the velocity uðτÞ prior to the acceleration. Analysis of the quadrature signals aðtÞ, bðtÞ, a0 ðtÞ, and b0 ðtÞ is explored in detail in [8,9,27]. However, these physics and the optical model show that leftcircular (ℓ) polarized light (see Appendix A), with relative phase difference presented in Eq. (7), interferes with h-polarized light when the early and late CW laser beams mix at the BS. As a simplistic explanation of the analysis, consider aðtÞ the cosine, and bðtÞ the sine, of the relative phase difference. If the vector defined by the amplitudes aðtÞ and bðtÞ is plotted versus time (the phasor), what is observed is that, before the reflector is in motion, the phasor
will noisily fluctuate around a fixed point (phase angle ϕ0 [28]) until the reflector begins to move, i.e., ϕ0 ¼ const and is the constant of integration that must be measured prior to the reflector motion if the surface velocity is to be determined after surface motion begins. Once the reflector begins to move, each time the phasor rotates 2πrad, the velocity has increased by the velocity per fringe constant of K ¼ λ0 =2Δt. Essentially, the difference velocity over time Δt along the path length to the BS must be integrated, i.e., if ω1 ≠ ω2 the number of oscillations of one frequency relative to the other will be different and must be counted. The integration of the difference phase occurs naturally in the interferometer as the Doppler-shifted frequencies traverse the optical paths to the final detector, as seen in Eqs. (7) and (8), where the velocity difference is multiplied by the path length x1 from the reflector to the BS divided by c the photon group velocity. The realization is that when the velocity of the reflector is constant, then the phase detected at the final BS remains constant. This last part is important because it demonstrates that, if ϕ0 is not measured prior to acceleration of the sample, then the velocity of the flyer cannot be known with VISAR-like systems; in contrast, heterodyne velocimetry systems determine the velocity of the system even when the measurement is performed on an object already in motion because the Doppler-shifted signal is heterodyned against the reference beam. Therefore, VISAR-like systems are at once heterodyne accelerometers and homodyne velocimeters, as shown in Eq. (7). In the homodyne case, the frequencies of the scattered light are constant over time, and the velocity undetermined by means of unreferenced difference interferometry techniques. In the heterodyne case, all rotations of the phasor are caused by accelerations of the sample—in the absence of accelerations no displacement is detected. Careful consideration of the physics shows that the autocompensation of the environmental fluctuations by the orthoconjugating Faraday rotator and reflector automatically corrects phase and polarization drifts caused by temperature and vibration. That is, relative phase and polarization drifts caused by length variations between the upper and the lower arms of the uMZ, and in the optical channel to and from the transmission and analysis optic, are automatically compensated for. In contrast to unbalanced VISAR systems, which use multimode fibers and lasers, because the proposed system is implemented with single-mode telecom components, neither will there be modal (polarization, transverse, or longitudinal) dispersion noise that appears simi-
^ (left image) composed of the Fig. 7. Polarization operator P three polarization optics (right image) labeled Q (quarter-wave retarder), F (22.5° Faraday rotator), and H (half-wave retarder).
Fig. 8. Upper half of each figure shows the polarization effects of the operator elements as light transmits from left to right and the lower half as light transmits from right-to-left after the light reflects from the shocked sample back into the optical channel ^ (b) the operation and into the uMZ: (a) describes the operation of P, of H, and (c) the operation of F when combined with the reflection from the surface of the shocked sample.
lar in character to speckle noise. Essentially, unless the light that reflects and couples back into the collecting fiber has noise from decoherent or coherent surface scattering processes, there will be no additional noise added to the single-mode fiber selfcorrecting system we propose. On the other hand, for the common VISAR, noise from coherent or decoherent scattering processes amplifies the observed effects of the noise on fringe visibility because of the modal dispersions along the multiple path lengths the light travels to the interfering BS. The implication is that the autocompensated system will be much less sensitive to decoherent and coherent surface scattering processes, leading to noise stabilization of ϕ0 and higher fringe visibility. 3. Conclusions
We established the physics of a single-mode, selfstabilized and passively balanced optical heterodyne accelerometer that can be assembled with standard telecom fiber components so that it has, essentially, zero longitudinal, transverse, and polarization mode dispersion. Further, the interfered intensities are fully balanced by the passive stabilization of the interferometer with an orthoconjugating technique. The final realization is that the inclusion of passive balancing and compensation reduces the path length difference of the uMZ to 1 μs, when Δt ¼ 1 ns, and 10 August 2010 / Vol. 49, No. 23 / APPLIED OPTICS
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when the shocked surface is traveling 1 mm=μs, which reduces the temporal decoherence term (that associates the length difference of the upper and lower arms of the uMZ) relative to the VISAR decoherence term by c=huðτÞi in time Δt. Because the system uses standard optical communication components, it can be fielded coincidentally with fiber-based LDV systems accessing the same transmission and reflection optical channel. It can also be modified to include additional fringe constants. While we have not tested the concept, a system is presently under development. Erskine and Holmes [29] proposed a similar “white light” velocimeter in 1995. In their case, they used a laser source through a uMZ that reflected from the shocked surface and recollected the light and directed it back to the uMZ. They did not include the F, ^ in their system, which resulted the PBS, or the P in a loss of correlated intensity at the BS (as shown near the detectors in Fig. 2), and an inability to perform quadrature analysis. The loss of correlated intensity caused an increase in the background noise in the analysis, reducing the fringe visibility of a weakly coherent system to begin with. Appendix A
^ seen in Fig. 4 is defined The polarization operator P as shown in Fig. 7. When h-polarized light is incident left to right, the optical elements are aligned to transmit h unchanged, but when h-polarized light is incident from right to left, the alignment of the elements rotates h to left-circular polarization, ℓ. This operational mode is achieved by aligning the fast and slow axes of Q along the h and v polarizations, so that when h is incident from left to right, it transmits unchanged past Q. Next, the 22:5° Faraday rotator F rotates h by 22:5°, and then H has its fast and slow axes aligned to rotate the 22:5° polarized light back to h for transmission to the right beyond PBS2 . Because the direction of polarization rotations caused by F is independent of the direction of incidence and because rotations caused by half-wave retarders are sensitive to the direction of incidence of light, on return, the operations of H followed by F rotates the h to a 45° diagonal polarization relative to the fast and slow axes of Q, diagonal polarization to rotate to circular polarization. The result is that on return, h-polarized light is interfered with ℓ-polarized light at the final BS, enabling quadrature detection of the interfering signals. Figure 8 shows how each of the polarizing elements alters the polarization of the light as it propagates through the system. Figure 8(a) describes the ^ in the short arm of the uMZ, Fig. 8(b) effect of P describes the effect of the λ=2 wave plate in the upper arm of the uMZ, and Fig. 8(c) describes the combined effects of the 45° Faraday rotator F and the reflection from the shocked sample. The interested reader is referred to the quantum-cryptography work of Bethune and Risk [22] for other discussions of the physics of orthoconjugating geometry. 4432
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References and Notes 1. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958). 2. H. Cummins, N. Knable, L. Gampel, and Y. Yeh, “Frequency shifts in light diffracted by ultrasonic waves in liquid media,” Appl. Phys. Lett. 2, 62–64 (1963). 3. H. Z. Cummins, N. Knable, and Y. Yeh, “Spurious harmonic generation in optical heterodyning,” Appl. Opt. 2, 823–825 (1963). 4. Y. Yeh and H. Z. Cummins, “Localized fluid flow measurements with an He─Ne laser,” Appl. Phys. Lett. 4, 176–178 (1964). 5. J. W. Forman, Jr., E. W. George, and R. D. Lewis, “Measurement of localized fluid flow velocities in gasses with a laser Doppler flowmeter,” Appl. Phys. Lett. 7, 77–78 (1965). 6. L. M. Barker and R. E. Hollenbach, “Interferometry technique for measuring the dynamic mechanical properties of materials,” Rev. Sci. Instrum. 36, 1617–1620 (1965). 7. L. M. Barker, Behavior of Dense Media under High Dynamic Pressures (Gordon & Breach, 1968), p. 483. 8. L. M. Barker and Hollenbach, “Laser interferometry for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669–4675 (1972). 9. W. F. Hemsing, “Velocity sensing interferometer (VISAR) modification,” Rev. Mod. Instrum. 50, 73–78 (1979). 10. L. Levin, D. Tzach, and J. Shamir, “Fiber optic velocity interferometer with very short coherence length light source,” Rev. Sci. Instrum. 67, 1434–1437 (1996). 11. L. Fabiny and A. D. Kersey, “Interferometric fiber-optic Doppler velocimeter with high-dynamic range,” IEEE Photon. Technol. Lett. 9, 79–81 (1997). 12. W. T. Buttler, S. K. Lamoreaux, F. G. Omenetto, and J. R. Torgerson, “Optical velocimetry,” arXiv:0409073v1 (2004). 13. O. T. Strand, D. R. Goosman, C. Martinex, and R. L. Whitworth, “Compact system for high-speed velocimetry using heterodyne techniques,” Rev. Sci. Instrum. 77, 083108 (2006). 14. “Quickly” relates to the temporal length of the delay leg, the laser wavelength, the reflector velocity, and the detection and recording system. 15. W. T. Buttler, “Comment on ‘accuracy limits and window corrections for photon Doppler velocimetry’ [J. Appl. Phys. 101, 013523 (2007)],” J. Appl. Phys. 103, 046102 (2008). 16. C. H. Bennett, “Quantum cryptography using any 2 nonorthogonal states,” Phys. Rev. Lett. 68, 3121–3124 (1992). 17. P. Townsend, J. G. Rarity, and P. Tapster, “Single photon interference in 10 km long optical fibre interferometer,” Electron. Lett. 29, 634–635 (1993). 18. P. Townsend, J. G. Rarity, and P. Tapster, “Enhanced single photon fringe visibility in a 10 km-long prototype quantum cryptography channel,” Electron. Lett. 29, 1291–1293 (1993). 19. M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989). 20. A. Muller, J. Bréguet, and N. Gisin, “Experimental demonstration of quantum cryptography using polarized photons in optical-fiber over more than 1 km,” Europhys. Lett. 23, 383–388 (1993). 21. J. Bréguet, A. Muller, and N. Gisin, “Quantum cryptography with polarized photons in optical fiber—experiment and practical limits,” J. Mod. Opt. 41, 2405–2412 (1994). 22. D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 42 (2002). 23. This operator, described in Appendix A, transmits h-polarized light traveling from left-to-right unchanged, but when the
reflected upper path light returns, it rotates h to left-circular (ℓ) polarization for quadrature detection. 24. What we refer to here is a combination of the shocked surface (reflector) with a Faraday rotator. If the reflector is a mirror, this combination is known as a Faraday mirror: it rotates reflected polarizations by 90° (orthoconjugates). 25. G. M. B. Bouricius and S. F. Clifford, “An optical interferometer using polarization coding to obtain quadrature phase components,” Rev. Sci. Instrum. 41, 1800–1803 (1970).
26. The δt terms are small enough to be ignored because Δx ≈ ½uðτÞ þ uðτ þ ΔtÞ=cΔt demonstrating that Δt · ω0 ½u2 ðτ þ Δt0 Þ− u2 ðτÞ=c2 ≪ 1. 27. D. H. Dolan, “Foundations of VISAR analysis,” Internal report SAND2006-1950 (Sandia National Laboratories, 2006). 28. This constant fluctuates as a function of time due to environmental factors. 29. D. J. Erskine and N. C. Holmes, “White light velocimetry,” Nature 377, 317–320 (1995).
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