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], optical heterodyne velocimetry techniques quickly followed as early practical uses of lasers [2, 3, 4, 5]. These early optical/laser Doppler velocimetry (LDV) techniques typically included simple optics, such as balanced 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, 3], Barker and Hollenbach demonstrated LDV normally reflected from materials under uniaxial strain [6], a demonstration that determined particle velocities of the order of 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]. 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, 9, 10, 11].
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 ), 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 , 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 , 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 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, 17, 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, 21, 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 telecom-fiber- laser that transmits CW horizontally polarized light (h), optical circulator (C), three polarizing beam splitters (PBSs), single BS, half-wave retarder (), polarization operator () [23], Faraday rotator (F) [24], a flyer plate or dynamic reflector (shocked sample), and four detectors (Det) used in pairs for quadrature detection [25] of the interference am plitudes ; 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 . The light then reflects from the shocked sample back along the optical path and is rotated an additional 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 in the uMZ upper arm, and the F with the shocked sample are described in Appendix A.
2. Physics and Optical Model
Consider , a horizontally polarized laser beam traveling left to right between the laser and the BS entrance to the uMZ:
where is the frequency of the laser light, and 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 is the laser-light correlation function, for which the light is assumed to be correlated over times much greater than , the temporal difference between the upper and the lower arms of the uMZ. As (h-polarized light) transmits from left to right, the circulator C transmits h directly through toward the BS that divides , 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 rotates the h to v, while in the lower path, transmits h unchanged, so that after the beams in the upper and lower paths emerge from the uMZ past , : In this arrangement of optical elements, v-polarized lags h-polarized by .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 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 , we have light in the polarization state of :
where now v-polarized light is leading h-polarized light because the F has rotated the polarizations by a total of . At this point it is clear that will reflect into the upper path, and 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 . 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 , when the sample is in motion.
With these physics we can determine the relative phase at the BS:
where, under the assumption of normal incidence and reflection, the Doppler-shifted frequencies are where . With these definitions, the relative phase difference at the BS, which is the quantity determined as a function of time through quadrature analysis, can be shown to be Equation (6) can be rewritten as if terms that scale as are ignored [26], demonstrating that relative to the initial phase when ( is a constant needed for integration).When , Eq. (7) can be rearranged to show that
where is the acceleration of the surface over time , , and where the velocity per fringe constant relates the increase in velocity of the surface relative to the velocity prior to the acceleration.Analysis of the quadrature signals , , , and is explored in detail in [8, 9, 27]. However, these physics and the optical model show that left-circular (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 the cosine, and the sine, of the relative phase difference. If the vector defined by the amplitudes and 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 [28]) until the reflector begins to move, i.e., 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 rad, the velocity has increased by the velocity per fringe constant of . Essentially, the difference velocity over time along the path length to the BS must be integrated, i.e., if 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 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 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 similar 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 self- correcting 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 and higher fringe visibility.
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 , when , and when the shocked surface is traveling , 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 in time . 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, the PBS, or the 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 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, 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 Faraday rotator rotates h by , and then H has its fast and slow axes aligned to rotate the polarized light back to h for transmission to the right beyond . Because the direction of polarization rotations caused by 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 rotates the h to a 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 8a describes the effect of in the short arm of the uMZ, Fig. 8b describes the effect of the wave plate in the upper arm of the uMZ, and Fig. 8c describes the combined effects of the 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.
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