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Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 49, Iss. 23 — Aug. 10, 2010
  • pp: 4427–4433
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Optical heterodyne accelerometry: passively stabilized, fully balanced velocity interferometer system for any reflector

William T. Buttler and Steven K. Lamoreaux  »View Author Affiliations


Applied Optics, Vol. 49, Issue 23, pp. 4427-4433 (2010)
http://dx.doi.org/10.1364/AO.49.004427


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Abstract

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

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

1. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958). [CrossRef]

], optical heterodyne velocimetry techniques quickly followed as early practical uses of lasers [2

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). [CrossRef]

, 3

3. H. Z. Cummins, N. Knable, and Y. Yeh, “Spurious harmonic generation in optical heterodyning,” Appl. Opt. 2, 823–825 (1963). [CrossRef]

, 4

4. Y. Yeh and H. Z. Cummins, “Localized fluid flow measurements with an HeNe laser,” Appl. Phys. Lett. 4, 176–178 (1964). [CrossRef]

, 5

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). [CrossRef]

]. 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 inter ferometer and a photomultiplier tube to monitor the interference beat frequency.

Within two years of the early Doppler work [2

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). [CrossRef]

, 3

3. H. Z. Cummins, N. Knable, and Y. Yeh, “Spurious harmonic generation in optical heterodyning,” Appl. Opt. 2, 823–825 (1963). [CrossRef]

], Barker and Hollenbach demonstrated LDV normally reflected from materials under uniaxial strain [6

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). [CrossRef]

], a demonstration that determined particle velocities of the order of 102m/s with a Michelson interferometer and fast (for the time) photodetectors. The implication of this result for the advancement of shock phys ics 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 un balanced Mach–Zehnder (uMZ) interferometer similar to that seen in Fig. 1 [7

7. L. M. Barker, Behavior of Dense Media under High Dynamic Pressures (Gordon & Breach, 1968), p. 483.

]. The realization of the apparatus formed an optical heterodyne accelerometer that superposes (interferes) early Doppler-shifted 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

8. L. M. Barker, “Laser interferometry for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669–4675 (1972). [CrossRef]

, 9

9. W. F. Hemsing, “Velocity sensing interferometer (VISAR) modification,” Rev. Mod. Instrum. 50, 73–78 (1979). [CrossRef]

, 10

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). [CrossRef]

, 11

11. L. Fabiny and A. D. Kersey, “Interferometric fiber-optic Doppler velocimeter with high-dynamic range,” IEEE Photon. Technol. Lett. 9, 79–81 (1997). [CrossRef]

].

With the development of optical communication laser systems, which incorporate high-power long- wavelength laser systems (e.g., single frequency and single spatial mode fiber-laser systems with λ=1550nm), high-bandwidth infrared detectors and oscilloscopes (recording systems), direct optical Doppler measurements of high velocities from materials under high strain are now possible [12

12. W. T. Buttler, S. K. Lamoreaux, F. G. Omenetto, and J. R. Torgerson, “Optical velocimetry,” arXiv:0409073v1 (2004).

, 13

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). [CrossRef]

]. However, while these systems have been demonstrated to directly determine velocities up to a few 103m/s, they have difficulty resolving slower velocities attributable to elastic precursors of shocked materials in the presence of sharply increasing veloc ities. 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 104m/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

14. “Quickly” relates to the temporal length of the delay leg, the laser wavelength, the reflector velocity, and the detection and recording system.

]. 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

15. W. T. Buttler, “Comment on ‘accuracy limits and window corrections for photon Doppler velocimetry’ [J. Appl. Phys. 101, 013523 (2007)] [CrossRef] [CrossRef] ,” J. Appl. Phys. 103, 046102 (2008).

].

In this paper, we present a VISAR-like velocity differencing interferometer that has its length, field, and intensity modes passively stabilized with an interferometric concept developed for use in fiber-based quantum cryptography. For example, early fiber-based quantum-cryptography concepts linked two uMZs by single-mode optical fibers [16

16. C. H. Bennett, “Quantum cryptography using any 2 nonorthogonal states,” Phys. Rev. Lett. 68, 3121–3124 (1992). [CrossRef] [PubMed]

, 17

17. P. Townsend, J. G. Rarity, and P. Tapster, “Single photon interference in 10km long optical fibre interferometer,” Electron. Lett. 29, 634–635 (1993). [CrossRef]

, 18

18. P. Townsend, J. G. Rarity, and P. Tapster, “Enhanced single photon fringe visibility in a 10km-long prototype quantum cryptography channel,” Electron. Lett. 29, 1291–1293 (1993). [CrossRef]

], 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

19. M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989). [CrossRef]

] system that is known as plug-and-play (autocompensating) quantum cryptography [20

20. A. Muller, J. Bréguet, and N. Gisin, “Experimental demonstration of quantum cryptography using polarized photons in optical-fiber over more than 1km,” Europhys. Lett. 23, 383–388 (1993). [CrossRef]

, 21

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). [CrossRef]

, 22

22. D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 42 (2002). [CrossRef]

], 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 1550nm telecom-fiber- laser that transmits CW horizontally polarized light (h), optical circulator (C), three polarizing beam splitters (PBSs), single BS, 90° half-wave retarder (λ/2), polarization operator (P^) [23

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 () polarization for quadrature detection.

], 45° Faraday rotator (F) [24

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).

], a flyer plate or dynamic reflector (shocked sample), and four detectors (Det) used in pairs for quadrature detection [25

25. G. M. B. Bouricius and S. F. Clifford, “An optical inter ferometer using polarization coding to obtain quadrature phase components,” Rev. Sci. Instrum. 41, 1800–1803 (1970). [CrossRef]

] of the interference am plitudes a(t),b(t),a(t),b(t); 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 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 P^ in the uMZ short arm, the λ/2 in the uMZ upper arm, and the F with the shocked sample are described in Appendix A.

2. Physics and Optical Model

Consider ψ0, a horizontally polarized laser beam traveling left to right between the laser and the BS entrance to the uMZ:
ψ0=A0[10]g(τ;ω0)=A0[h]g(τ;ω0),
(1)
where ω0 is the frequency of the laser light, and τ=t+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 g(τ) is the laser-light correlation function, for which the light is assumed to be correlated over times much greater than Δt=2L/c, the temporal difference between the upper and the lower arms of the uMZ. As ψ0 (h-polarized light) transmits from left to right, the circulator C transmits h directly through PBS1 toward the BS that divides ψ0, 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 rotates the h to v, while in the lower path, P^ transmits h unchanged, so that after the beams in the upper and lower paths emerge from the uMZ past PBS2, ψ0ψ1+ψ2=ψ:
ψ1=A02[10]g(τ;ω0)=A02[h]g(τ;ω0),ψ2=A02[01]g(τ+Δt;ω0)=A02[v]g(τ+Δt;ω0).
(2)
In this arrangement of optical elements, v-polarized ψ2 lags h-polarized ψ1 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=ψ1+ψ2:
ψ=A02([v]g(τ;ω0)+[h]g(τ+Δt;ω0)),
(3)
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 ψ1 will reflect into the upper path, and ψ2 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.

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 sub jected 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 high- explosive (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 1mm/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 P12Δx=2(x1+LΔx)=P2, when the sample is in motion.

With these physics we can determine the relative phase at the BS:
ϕ1=ω0x12Δxc+ω1x1+2Lc,ϕ2=ω0x1+2LΔxc+ω2x1Δxc,
(4)
where, under the assumption of normal incidence and reflection, the Doppler-shifted frequencies are
ω1ωD(τ)=ω0(1+2u(τ)c),ω2ωD(τ+Δt)=ω0(1+2u(τ+Δt)c),
(5)
where Δt=Δt+δt=2L/c+Δx/c. With these definitions, the relative phase difference Δϕ at the BS, which is the quantity determined as a function of time through quadrature analysis, can be shown to be
Δϕ=(ω0ω1)2Lc+(ω0ω2)Δxc+(ω2ω1)x1c.
(6)
Equation (6) can be rewritten as
Δϕ2={ϕ0if    u1=u2=0ω0u1cΔtif    u1=u20ω0(u1cΔtu2u1cx1c)if   u1u2,
(7)
if terms that scale as δt=Δx/c are ignored [26

26. The δt terms are small enough to be ignored because Δx[u(τ)+u(τ+Δt)]/cΔt demonstrating that Δt·ω0[u2(τ+Δt)u2(τ)]/c21.

], demonstrating that Δϕ=const relative to the initial phase ϕ0 when u1=u2 (ϕ0 is a constant needed for integration).

When u1u2, Eq. (7) can be rearranged to show that
u(τ+Δt)u(τ)ΔtTa(τ)T=u(τ)+KΔϕ2π,
(8)
where a(τ) is the acceleration of the surface over time Δt, T=x1/c, and where the velocity per fringe constant K=λ0/2Δt relates the increase in velocity of the surface relative to the velocity u(τ) prior to the acceleration.

Analysis of the quadrature signals a(t), b(t), a(t), and b(t) is explored in detail in [8

8. L. M. Barker, “Laser interferometry for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669–4675 (1972). [CrossRef]

, 9

9. W. F. Hemsing, “Velocity sensing interferometer (VISAR) modification,” Rev. Mod. Instrum. 50, 73–78 (1979). [CrossRef]

, 27

27. D. H. Dolan, “Foundations of VISAR analysis,” Internal report SAND2006-1950 (Sandia National Laboratories, 2006).

]. However, these physics and the optical model show that left-circular () 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 a(t) the cosine, and b(t) the sine, of the relative phase difference. If the vector defined by the amplitudes a(t) and b(t) 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

28. This constant fluctuates as a function of time due to environmental factors.

]) until the reflector begins to move, i.e., ϕ0=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=λ0/2Δt. Essentially, the difference velocity over time Δt along the path length to the BS must be integrated, i.e., if ω1ω2 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, 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.

3. Conclusions

We established the physics of a single-mode, self- stabilized 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 1μs, when Δt=1ns, and when the shocked surface is traveling 1mm/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/u(τ) 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

29. D. J. Erskine and N. C. Holmes, “White light velocimetry,” Nature 377, 317–320 (1995). [CrossRef]

] 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, the PBS, or the P^ in their system, which resulted 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

The polarization operator P^ seen in Fig. 4 is defined 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, . 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 -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 8a describes the effect of P^ in the short arm of the uMZ, Fig. 8b describes the effect of the λ/2 wave plate in the upper arm of the uMZ, and Fig. 8c 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

22. D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 42 (2002). [CrossRef]

] for other discussions of the physics of orthoconjugating geometry.

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.)
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.
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.
Fig. 4 These elements define an autocompensating telecom- fiber-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 double-ended arrows, and optical paths where light travels in only one direction are marked with single-ended arrows.
Fig. 5 Notional experimental geometry that requires an ability to determine the velocity of a material sample that has experienced a high-pressure impulse—shock wave.
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.)
Fig. 7 Polarization operator P^ (left image) composed of the 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 and into the uMZ: (a) describes the operation of P^, (b) the operation of H, and (c) the operation of F when combined with the reflection from the surface of the shocked sample.
1.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958). [CrossRef]

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). [CrossRef]

3.

H. Z. Cummins, N. Knable, and Y. Yeh, “Spurious harmonic generation in optical heterodyning,” Appl. Opt. 2, 823–825 (1963). [CrossRef]

4.

Y. Yeh and H. Z. Cummins, “Localized fluid flow measurements with an HeNe laser,” Appl. Phys. Lett. 4, 176–178 (1964). [CrossRef]

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). [CrossRef]

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). [CrossRef]

7.

L. M. Barker, Behavior of Dense Media under High Dynamic Pressures (Gordon & Breach, 1968), p. 483.

8.

L. M. Barker, “Laser interferometry for measuring high velocities of any reflecting surface,” J. Appl. Phys. 43, 4669–4675 (1972). [CrossRef]

9.

W. F. Hemsing, “Velocity sensing interferometer (VISAR) modification,” Rev. Mod. Instrum. 50, 73–78 (1979). [CrossRef]

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). [CrossRef]

11.

L. Fabiny and A. D. Kersey, “Interferometric fiber-optic Doppler velocimeter with high-dynamic range,” IEEE Photon. Technol. Lett. 9, 79–81 (1997). [CrossRef]

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). [CrossRef]

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)] [CrossRef] [CrossRef] ,” J. Appl. Phys. 103, 046102 (2008).

16.

C. H. Bennett, “Quantum cryptography using any 2 nonorthogonal states,” Phys. Rev. Lett. 68, 3121–3124 (1992). [CrossRef] [PubMed]

17.

P. Townsend, J. G. Rarity, and P. Tapster, “Single photon interference in 10km long optical fibre interferometer,” Electron. Lett. 29, 634–635 (1993). [CrossRef]

18.

P. Townsend, J. G. Rarity, and P. Tapster, “Enhanced single photon fringe visibility in a 10km-long prototype quantum cryptography channel,” Electron. Lett. 29, 1291–1293 (1993). [CrossRef]

19.

M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989). [CrossRef]

20.

A. Muller, J. Bréguet, and N. Gisin, “Experimental demonstration of quantum cryptography using polarized photons in optical-fiber over more than 1km,” Europhys. Lett. 23, 383–388 (1993). [CrossRef]

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). [CrossRef]

22.

D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 42 (2002). [CrossRef]

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 () 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 inter ferometer using polarization coding to obtain quadrature phase components,” Rev. Sci. Instrum. 41, 1800–1803 (1970). [CrossRef]

26.

The δt terms are small enough to be ignored because Δx[u(τ)+u(τ+Δt)]/cΔt demonstrating that Δt·ω0[u2(τ+Δt)u2(τ)]/c21.

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). [CrossRef]

OCIS Codes
(120.4640) Instrumentation, measurement, and metrology : Optical instruments
(120.7250) Instrumentation, measurement, and metrology : Velocimetry
(290.5880) Scattering : Scattering, rough surfaces

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: March 25, 2010
Revised Manuscript: June 11, 2010
Manuscript Accepted: July 10, 2010
Published: August 6, 2010

Citation
William T. Buttler and Steven K. Lamoreaux, "Optical heterodyne accelerometry: passively stabilized, fully balanced velocity interferometer system for any reflector," Appl. Opt. 49, 4427-4433 (2010)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-49-23-4427


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References

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  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). [CrossRef]
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  19. M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  22. D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 42 (2002). [CrossRef]
  23. This operator, described in Appendix , 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). [CrossRef]
  26. The δt terms are small enough to be ignored because Δx≈[u(τ)+u(τ+Δt)]/cΔt demonstrating that Δt·ω0[u2(τ+Δt′)−u2(τ)]/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). [CrossRef]

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