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NASA NTRS2010

A Transportable Gravity Gradiometer Based on Atom Interferometry

A transportable atom interferometer-based gravity gradiometer has been developed at JPL to carry out measurements of Earth's gravity field at ever finer spatial resolutions, and to facilitate high-resolution monitoring of temporal variations in the gravity field from ground- and flight-based platforms. Existing satellite-based gravity missions such as CHAMP and GRACE measure the gravity field via precise monitoring of the motion of the satellites; i.e. the satellites themselves function as test masses. JPL's quantum gravity gradiometer employs a quantum phase measurement technique, similar to that employed in atomic clocks, made possible by recent advances in laser cooling and manipulation of atoms. This measurement technique is based on atomwave interferometry, and individual laser-cooled atoms are used as drag-free test masses. The quantum gravity gradiometer employs two identical atom interferometers as precision accelerometers to measure the difference in gravitational acceleration between two points (Figure 1). By using the same lasers for the manipulation of atoms in both interferometers, the accelerometers have a common reference frame and non-inertial accelerations are effectively rejected as common mode noise in the differential measurement of the gravity gradient. As a result, the dual atom interferometer-based gravity gradiometer allows gravity measurements on a moving platform, while achieving the same long-term stability of the best atomic clocks. In the laboratory-based prototype (Figure 2), the cesium atoms used in each atom interferometer are initially collected and cooled in two separate magneto-optic traps (MOTs). Each MOT, consisting of three orthogonal pairs of counter-propagating laser beams centered on a quadrupole magnetic field, collects up to 10(exp 9) atoms. These atoms are then launched vertically as in an atom fountain by switching off the magnetic field and introducing a slight frequency shift between pairs of lasers to create a moving rest frame for the trapped atoms. While still in this moving-frame molasses, the laser frequencies are further detuned from the atomic resonance (while maintaining this relative frequency shift) to cool the atom cloud's temperature to 2 K or below, corresponding to an rms velocity of less than 2 cm/s. After launch, the cold atoms undergo further state and velocity selection to prepare for atom interferometry. The atom interferometers are then realized using laser-induced stimulated Raman transitions to perform the necessary manipulations of each atom, and the resulting interferometer phase is measured using laser-induced fluorescence for state-normalized detection. More than 20 laser beams with independent controls of frequency, phase, and intensity are required for this measurement sequence. This instrument can facilitate the study of Earth's gravitational field from surface and air vehicles, as well as from space by allowing gravity mapping from a low-cost, single spacecraft mission. In addition, the operation of atom interferometer-based instruments in space offers greater sensitivity than is possible in terrestrial instruments due to the much longer interrogation times available in the microgravity environment. A space-based quantum gravity gradiometer has the potential to achieve sensitivities similar to the GRACE mission at long spatial wavelengths, and will also have resolution similar to GOCE for measurement at shorter length scales.

Yu, Nan↗

NASA NTRS2002

SIM white light on-board processing algorithms

Interferometry in optical astronomy is an important and growing field of astronomical observation. A number of stellar interferometers have come online over the past several years, and several more are due to be operational in the near future. In addition, space based interferometers are also planned missions of NASA's Origins program, including the Space Interferometry Mission (SIM), the focus of the present paper. The fundamental measurement made by each of these interferometers is the white light fringe measurement to determine the optical pathlength delay between the two arms of the interferometer. SIM makes white light measurements with three independent interferometers observing three different objects. Two of these are the "guide" interferometers that observe bright objects (approximately 7th magnitude) to track the rigid body motion of the instrument. The third interferometer, the "science" interferometer, observes the science targets of interest.

interferometry↗

NASA NTRS2002

Stellar Interferometry from the Ground and Space

Stellar Interferometry began more than 80 years ago with the pioneering measurement of the diameter of Betelqueuse by Michelson and Pease using a 20 foot beam mounted at the top of the 10011 Hooker telescope at Mt. Wilson. Essentially no other work was done in this field until the 1960's when Hanbury-Brown and his colleagues developed and used the Intensity Interferometer at Narrabri, Australia to measure the diameters of a number of important hot stars. The modern period of Stellar Interferometry really began in the 1970's with the successes of 3 or 4 small research groups in the US and Europe, and scientific and technical progress in the field has been outstanding, particularly in the last decade. This has lead to the development of two major ground based facilities: NASA's own Keck Interferometer and ESO's Very Large Telescope Interferometer, and a number of space interferometers such as the Space Interferometer Mission (SIM), and the Terrestrial Planet Finder (TPF), among others. I will review the principles, history, and scientific progress in the field both on the ground and in space, and I will discuss a mission concept under development here at NASA Goddard, the Fourier-Kelvin Stellar Interferometer, a near-term mid-infrared imaging interferometer, which can serve as a scientific and technical pre-cursor for some of the more ambitious concepts being discussed within the Astronomical and NASA communities.

Danchi, William C.↗

NASA NTRS2011

Multi-Segment Radius Measurement Using an Absolute Distance Meter Through a Null Assembly

This system was one of the test methods considered for measuring the radius of curvature of one or more of the 18 segmented mirrors that form the 6.5 m diameter primary mirror (PM) of the James Webb Space Telescope (JWST). The assembled telescope will be tested at cryogenic temperatures in a 17-m diameter by 27-m high vacuum chamber at the Johnson Space Center. This system uses a Leica Absolute Distance Meter (ADM), at a wavelength of 780 nm, combined with beam-steering and beam-shaping optics to make a differential distance measurement between a ring mirror on the reflective null assembly and individual PM segments. The ADM is located inside the same Pressure-Tight Enclosure (PTE) that houses the test interferometer. The PTE maintains the ADM and interferometer at ambient temperature and pressure so that they are not directly exposed to the telescope s harsh cryogenic and vacuum environment. This system takes advantage of the existing achromatic objective and reflective null assembly used by the test interferometer to direct four ADM beamlets to four PM segments through an optical path that is coincident with the interferometer beam. A mask, positioned on a linear slide, contains an array of 1.25 mm diameter circular subapertures that map to each of the 18 PM segments as well as six positions around the ring mirror. A down-collimated 4 mm ADM beam simultaneously covers 4 adjacent PM segment beamlets and one ring mirror beamlet. The radius, or spacing, of all 18 segments can be measured with the addition of two orthogonally-oriented scanning pentaprisms used to steer the ADM beam to any one of six different sub-aperture configurations at the plane of the ring mirror. The interferometer beam, at a wavelength of 687 nm, and the ADM beamlets, at a wavelength of 780 nm, pass through the objective and null so that the rays are normally incident on the parabolic PM surface. After reflecting off the PM, both the ADM and interferometer beams return to their respective instruments on nearly the same path. A fifth beamlet, acting as a differential reference, reflects off a ring mirror attached to the objective and null and returns to the ADM. The spacings between the ring mirror, objective, and null are known through manufacturing tolerances as well as through an in situ null wavefront alignment of the interferometer test beam with a reflective hologram located near the caustic of the null. Since total path length between the ring mirror and PM segments is highly deterministic, any ADM-measured departures from the predicted path length can be attributed to either spacing error or radius error in the PM. It is estimated that the path length measurement between the ring mirror and a PM segment is accurate to better than 100 m. The unique features of this invention include the differential distance measuring capability and its integration into an existing cryogenic and vacuum compatible interferometric optical test.

Merle, Cormic↗

NASA NTRS2012

Measurement Via Optical Near-Nulling and Subaperture Stitching

A subaperture stitching interferometer system provides near-nulling of a subaperture wavefront reflected from an object of interest over a portion of a surface of the object. A variable optical element located in the radiation path adjustably provides near-nulling to facilitate stitching of subaperture interferograms, creating an interferogram representative of the entire surface of interest. This enables testing of aspheric surfaces without null optics customized for each surface prescription. The surface shapes of objects such as lenses and other precision components are often measured with interferometry. However, interferometers have a limited capture range, and thus the test wavefront cannot be too different from the reference or the interference cannot be analyzed. Furthermore, the performance of the interferometer is usually best when the test and reference wavefronts are nearly identical (referred to as a null condition). Thus, it is necessary when performing such measurements to correct for known variations in shape to ensure that unintended variations are within the capture range of the interferometer and accurately measured. This invention is a system for nearnulling within a subaperture stitching interferometer, although in principle, the concept can be employed by wavefront measuring gauges other than interferometers. The system employs a light source for providing coherent radiation of a subaperture extent. An object of interest is placed to modify the radiation (e.g., to reflect or pass the radiation), and a variable optical element is located to interact with, and nearly null, the affected radiation. A detector or imaging device is situated to obtain interference patterns in the modified radiation. Multiple subaperture interferograms are taken and are stitched, or joined, to provide an interferogram representative of the entire surface of the object of interest. The primary aspect of the invention is the use of adjustable corrective optics in the context of subaperture stitching near-nulling interferometry, wherein a complex surface is analyzed via multiple, separate, overlapping interferograms. For complex surfaces, the problem of managing the identification and placement of corrective optics becomes even more pronounced, to the extent that in most cases the null corrector optics are specific to the particular asphere prescription and no others (i.e. another asphere requires completely different null correction optics). In principle, the near-nulling technique does not require subaperture stitching at all. Building a near-null system that is practically useful relies on two key features: simplicity and universality. If the system is too complex, it will be difficult to calibrate and model its manufacturing errors, rendering it useless as a precision metrology tool and/or prohibitively expensive. If the system is not applicable to a wide range of test parts, then it does not provide significant value over conventional null-correction technology. Subaperture stitching enables simpler and more universal near-null systems to be effective, because a fraction of a surface is necessarily less complex than the whole surface (excepting the extreme case of a fractal surface description). The technique of near-nulling can significantly enhance aspheric subaperture stitching capability by allowing the interferometer to capture a wider range of aspheres. More over, subaperture stitching is essential to a truly effective near-nulling system, since looking at a fraction of the surface keeps the wavefront complexity within the capability of a relatively simple nearnull apparatus. Furthermore, by reducing the subaperture size, the complexity of the measured wavefront can be reduced until it is within the capability of the near-null design.

Forbes, Greg↗

NASA NTRS2002

SIM testbed 3 real-time control software

SIM System Testbed 3 (STB3) features three optical interferometers sharing a common baseline, as a dynamic representation of the SIM instrument. An artificial star feeding the interferometers is installed on a separate optics bench. All three interferometers use photons captured by avalanche photo diodes (APDs) to measure the position and quality of fringes, and additional pointing precision is achieved by fast steering mirrors (FSMs) that keep the star images centered on the beam combining optics using a CCD camera. Each interferometer uses internal metrology to measure changes in its optical pathlength. External metrology beams measure changes in the baseline vector. This system acquires and tracks white light fringes with one interferometer, while the other two acquire and track laser light fringes representing the bright guide stars that will be used by SIM. The white light source represents a dim star that cannot supply enough photons for the Science interferometer to lock onto fringes in closed-loop mode; instead it operates open-loop, using pathlength corrections fed to it from the two guide interferometers and the external metrology subsystem to reject disturbances and maintain the fringes. This tracking mode is known as Pathlength Feed Forward (PFF). The precise real-time behavior required to achieve this result is implemented by a complex set of interacting software control loops. This paper describes how these loops take advantage of the benefits of the RTC Core architecture, and how they work together to accomplish STB3's objectives.

interferometry↗

DOE OSTI2013

Talbot-Lau based Moiré deflectometry with non-coherent sources as potential High Energy Density plasma diagnostic

X-ray phase-contrast radiography could better characterize highly localized density gradients expected in High Energy Density (HED) plasma experiments than conventional attenuation radiography. In particular, the Talbot-Lau (TL) grating interferometer, which works with extended and polychromatic x-ray sources, is a potentially attractive HED diagnostic due to its high sensitivity. For HED characterization the TL setup and imaging techniques must be changed from the recently studied medical system. The object magnification must be increased greatly in order to resolve μm scale gradients while the Talbot magnification must be increased in order to keep the gratings away from the plasma. Additionally, techniques for retrieving the density profile from a single plasma image must be developed. We thus study the performance of high magnification TL interferometers, in conjunction with Moiré fringe deflectometry for single image phase retrieval. The results show a very good interferometer contrast (≤30%) at high magnification. The Moiré technique enables measuring both sharp and mild density gradients with good accuracy and spatial resolution. Both the laboratory and simulation studies indicate that the TL based Moiré deflectometry is more sensitive than the propagation phase-contrast method when utilizing an extended x-ray source (∼80 μm). In HED experiments this would allow for less demanding X-ray backlighters than those used at present.

36 MATERIALS SCIENCE↗

DOE OSTI2011

Atom-interferometric techniques for measuring uniform magnetic field gradients and gravitational acceleration

We discuss techniques for probing the effects of a constant force acting on cold atoms using two configurations of a grating echo-type atom interferometer. Laser-cooled samples of {sup 85}Rb with temperatures as low as 2.4 {mu}K have been achieved in a new experimental apparatus with a well-controlled magnetic environment. We demonstrate interferometer signal lifetimes approaching the transit time limit in this system ({approx}270 ms), which is comparable to the time scale achieved by Raman interferometers. Using these long time scales, we experimentally investigate the influence of a homogeneous magnetic field gradient using two- and three-pulse interferometers, which enable us to sense changes in externally applied magnetic field gradients as small as {approx}4x10{sup -5} G/cm. We also provide an improved theoretical description of signals generated by both interferometer configurations that accurately models experimental results. With this theory, absolute measurements of B gradients at the level of 3x10{sup -4} G/cm are achieved. Finally, we contrast the suitability of the two- and three-pulse interferometers for precision measurements of the gravitational acceleration, g.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2023

Coriolis force compensation and laser beam delivery for 100-m baseline atom interferometry

The Coriolis force is a significant source of systematic phase errors and dephasing in atom interferometry and is often compensated by counter-rotating the interferometry laser beam against Earth's rotation. We present a novel method for performing Coriolis force compensation for long-baseline atom interferometry, which mitigates atom-beam misalignment due to beam rotation, an effect which is magnified by the long lever arm of the baseline length. The method involves adjustment of the angle of the interferometer beam prior to a magnifying telescope, enabling the beam to pivot around a tunable position along the interferometer baseline. By tuning the initial atom kinematics and adjusting the angle with which the interferometer beam pivots about this point, we can ensure that the atoms align with the center of the beam during the atom optics laser pulses. This approach will be used in the MAGIS-100 atom interferometer and could also be applied to other long-baseline atom interferometers. An additional challenge associated with long baseline interferometry is that since long-baseline atom interferometers are often located outside of typical laboratory environments, facilities constraints may require lasers to be housed in a climate-controlled room a significant distance away from the main experiment. Nonlinear effects in optical fibers restrict the use of fiber-based transport of the high-power interferometry beam from the laser room to the experiment. We present the design of and prototype data from a laser transport system for MAGIS-100 that maintains robustness against alignment drifts despite the absence of a long fiber.

Glick, Jonah↗

NASA NTRS1991

Extra-solar planet detection

Extra-solar planet detection has been a goal of astronomers for many decades. This paper describes current efforts in planet detection using interferometric techniques. At present, the Mark III long baseline interferometer has been operational for a number of years. The Mark III has achieved significant improvement in astrometric accuracy in two areas, wide angle astrometry and double star astrometry. Two new interferometers are being developed. The first is a direct combination IR interferometer, an upgrade of the UCB IR heterodyne interferometer. The second is the Keck Interferometer Array. This instrument, to be operational at the end of the decade will be a major interferometric facility, with the capability to combine coherently the light from the two 10-meter Keck telescopes as well as four 1.5-meter movable outrigger telescopes. The last project directed at planet detection is OSI, a space-based long-baseline interferometer with a planned astrometric accuracy of 1-10 microarcsec.

Shao, Michael↗

NASA NTRS1993

Study of Einstein-Podolsky-Rosen state for space-time variables in a two photon interference experiment

A pair of correlated photons generated from parametric down conversion was sent to two independent Michelson interferometers. Second order interference was studied by means of a coincidence measurement between the outputs of two interferometers. The reported experiment and analysis studied this second order interference phenomena from the point of view of Einstein-Podolsky-Rosen paradox. The experiment was done in two steps. The first step of the experiment used 50 psec and 3 nsec coincidence time windows simultaneously. The 50 psec window was able to distinguish a 1.5 cm optical path difference in the interferometers. The interference visibility was measured to be 38 percent and 21 percent for the 50 psec time window and 22 percent and 7 percent for the 3 nsec time window, when the optical path difference of the interferometers were 2 cm and 4 cm, respectively. By comparing the visibilities between these two windows, the experiment showed the non-classical effect which resulted from an E.P.R. state. The second step of the experiment used a 20 psec coincidence time window, which was able to distinguish a 6 mm optical path difference in the interferometers. The interference visibilities were measured to be 59 percent for an optical path difference of 7 mm. This is the first observation of visibility greater than 50 percent for a two interferometer E.P.R. experiment which demonstrates nonclassical correlation of space-time variables.

Shih, Y. H.↗

NASA NTRS1998

Modulated Source Interferometry with Combined Amplitude and Frequency Modulation

An improved interferometer is produced by modifying a conventional interferometer to include amplitude and/or frequency modulation of a coherent light source at radio or higher frequencies. The phase of the modulation signal can be detected in an interfering beam from an interferometer and can be used to determine the actual optical phase of the beam. As such, this improvement can be adapted to virtually any two-beam interferometer, including: Michelson, Mach-Zehnder, and Sagnac interferometers. The use of an amplitude modulated coherent tight source results in an interferometer that combines the wide range advantages of coherent interferometry with the precise distance measurement advantages of white light interferometry.

Gutierrez, Roman C.↗

NASA NTRS2002

Detection, Evaluation, and Optimization of Optical Signals Generated by Fiber Optic Bragg Gratings Under Dynamic Excitations

The dynamic response of a fiber optic Bragg grating to mechanical vibrations is examined both theoretically and experimentally. The theoretical expressions describing the consequences of changes in the grating's reflection spectrum are derived for partially coherent beams in an interferometer. The analysis is given in terms of the dominant wavelength, optical bandwidth, and optical path difference of the interfering signals. Changes in the reflection spectrum caused by a periodic stretching and compression of the grating were experimentally measured using an unbalanced Michelson interferometer, a Michelson interferometer with a non-zero optical path difference. The interferometer's sensitivity to changes in dominant wavelength of the interfering beams was measured as a function of interferometer unbalance and was compared to theoretical predictions. The theoretical analysis enables the user to determine the optimum performance for an unbalanced interferometer.

Adamovsky, Grigory↗

NASA NTRS2001

Diffusivity Measurements Made Instant and Easy

A compact common path interferometer (CPI) system has been developed to measure the diffusivity of liquid pairs. The CPI is an optical technique that can be used to measure changes in the gradient of the refraction index of transparent materials. It uses a shearing interferometer that shares the same optical path from a laser light source to the final imaging plane. The molecular diffusion coefficient of liquids can be determined from the physical relations between changes in the optical path length and liquid phase properties. When the data obtained by using the CPI have been compared with similar results from other techniques, the instrument has been demonstrated to be far superior to other instruments for measuring the diffusivity of miscible liquids while staying very compact and robust (ref. 1). Because of its compactness and ease of use, the CPI has been adopted for use in studies of interface dynamics as well as other diffusion-controlled process applications (ref. 2). This progress will permit experiments in microgravity that can quantitatively answer basic science questions about mass and thermal diffusion and their effect in transport processes. This instrument is a spinoff of a diagnostic development for microgravity fluid physics experiments at the NASA Glenn Research Center that has used optics and electronics existing in the fluid physics laboratory for feasibility studies. Optical diagnostic techniques have become an integral part of many areas of measurement applications in industrial and research laboratories. Many types of interferometers and their phase-shifted versions have been used as instruments for measuring optical wave fronts for lens testing and combustion and fluid flow diagnostics. One of these, the point diffraction interferometer, is considered to be robust (see, for example, ref. 3) because it has a common-path design. The point diffraction interferometer is difficult to align and has a limited measurement range for liquid-phase applications. Interferometry and schlieren techniques have been widely used for many years for gas-flow-visualization. On the other hand, the CPI, which is basically a Wollaston prism Polarizer in combination with an analyser (crystalline quartz retardation plate) is used instead of a point diffraction interferometer unit. The advantage of using the CPI over other optical techniques is that it can make quantitative measurements in liquids with the large index of refraction variations that often occur in interface dynamics studies. This can be simply accomplished by using different prism polarizers for each particular experimental condition.

Rashidnia, Nasser↗

DOE OSTI2017

Theorem of comparative sensitivity of fibre sensors

We report an analysis of sensitivity of fibre sensors of physical quantities based on different types of interferometers. We formulate and prove the following theorem: under the time-dependent external physical perturbations at nonzero frequencies (i.e., except the static and low-frequency ones) on the sensitive arms of an interferometer in the form of multiturn elements (coils), there exist such lengths L of the measuring arms of the fibre interferometers at which the sensitivity of sensors based on the Sagnac fibre interferometers can be comparable with the sensitivity of sensors based on Michelson, Mach – Zehnder, or Fabry – Perot fibre interferometers, as well as exceed it under similar other conditions (similar-type perturbations, similar arm lengths and single-mode fibre types). The consequences that follow from the theorem, important for practical implementation of arrays of fibre sensors for measurement purposes and the devices with stable metrological properties, are discussed. (paper)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2004

Progress towards a space-borne quantum gravity gradiometer

Quantum interferometer gravity gradiometer for 3D mapping is a project for developing the technology of atom interferometer-based gravity sensor in space. The atom interferometer utilizes atomic particles as free fall test masses to measure inertial forces with unprecedented sensitivity and precision. It also allows measurements of the gravity gradient tensor components for 3D mapping of subsurface mass distribution. The overall approach is based on recent advances of laser cooling and manipulation of atoms in atomic and optical physics. Atom interferometers have been demonstrated in research laboratories for gravity and gravity gradient measurements. In this approach, atoms are first laser cooled to micro-kelvin temperatures. Then they are allowed to freefall in vacuum as true drag-free test masses. During the free fall, a sequence of laser pulses is used to split and recombine the atom waves to realize the interferometric measurements. We have demonstrated atom interferometer operation in the Phase I period, and we are implementing the second generation for a complete gradiometer demonstration unit in the laboratory. Along with this development, we are developing technologies at component levels that will be more suited for realization of a space instrument. We will present an update of these developments and discuss the future directions of the quantum gravity gradiometer project.

laser cooling↗

NASA NTRS2018

Direct Probe of Dark Energy Interactions with a Solar System Laboratory: NASA Innovative Advanced Concepts (NIAC) - Phase 1 Final Report

In this NIAC (NASA Innovative Advanced Concepts) study, we embrace the challenge of direct detection of the galileon dark energy field in the Vainshtein model. We developed a mission concept to directly measure the galileon field using the solar system as a laboratory. The experiment scheme involves precise measurements of the trace of the total scalar force gradient tensor. A tetrahedral constellation off our spacecraft measures the "local" traces while orbiting about 1 AU (Astronomical Unit) away from the Sun and faraway from planets (Figure 1). The trace measurement is insensitive to the much stronger gravity field which satisfies the inverse square law and thus is traceless. Atomic test masses and atom interferometer measurement techniques are used as precise drag-free inertial references while laser ranging interferometers are employed to connect among atom interferometer pairs in spacecraft for the differential gradient force measurements. We conclude that such a mission is scientifically and technologically feasible. We show that a mission of 3-year measurement time would be able to provide high confidence statements (over 3 standard deviations) about the existence and strength of the cubic galileon field of the Sun. In addition, such a mission would also provide rich and diverse scientific data for testing any gravitational theory in general beyond the Newtonian gravity, hunting for ultra-light fields of dark matter, and detecting gravitational waves in the mid-frequency band between those of LIGO (Laser Interferometer Gravitational-Wave Observatory) and LISA (Laser Interferometer Space Antenna). For these reasons, we will term the mission concept Gravity Observation and Dark energy Detection Explorer in the Solar System (GODDESS).

Dark Energy↗

DOE OSTI2026

Multiloop and multiaxis atomtronic Sagnac interferometry

We report the experimental realization of a large-area and multiaxis atomtronic interferometer in an optical waveguide for rotation sensing. A large enclosed area is achieved through multiloop operation in a guided-atom interferometer using Bose-Einstein condensates. We demonstrate a three-loop interferometer with a total interrogation time of ∼ 0.4 s and an enclosed area of 8.7 mm 2 —the largest reported in a fully guided or one-dimensional setup. High-contrast interference fringes are observed for up to five Sagnac orbits in a smaller loop-area configuration. Our approach enables interleaved rotation measurements about multiple arbitrary axes within the same experimental setup. We present results for area-enclosing interferometers in both horizontal and vertical planes, demonstrating that the interferometer contrast remains comparable across orthogonal orientations of the enclosed area vectors.

74 ATOMIC AND MOLECULAR PHYSICS↗