Interferometric Phase Measurements Research Articles

Measuring the transmitted wavefront of large-aperture flat optics

This paper proposes a scanning distance measuring interferometer (SDMI) method to measure the transmitted wavefront of large-aperture flat optics accurately. The biaxial positioning mechanism moves the test component through the interferometric path of a differential distance-measuring interferometer; both the optical path difference and distribution are obtained; thus, the wavefront can be reconstructed. A Ø360 mm flat optical substrate is measured, and the result is compared with a phase measuring interferometer (PMI). The SDMI result is 68 nm in PV and 13 nm in RMS. The PMI result is 90.8 nm in PV and 16.4 nm in RMS. The forms of the two results show a good correlation, which gives confidence in the viability of the SDMI method for large-aperture optical transmitted wavefront measurements out of range of the commercial PMI.

Analysis of edge jumps in multi-wavelength phasing of segmented-mirror telescopes.

Interferometric methods for phasing the primary mirrors of segmented-mirror telescopes generally need to make measurements at multiple wavelengths in order to achieve suitably large edge height capture ranges. In the presence of measurement errors, such methods have the potential for grossly wrong measurements, so-called edge jumps. In this paper, we investigate the causes of edge jumps using the example of Thirty Meter Telescope (TMT) phasing. We explain the reasons for their occurrence and develop a mathematical understanding of the conditions under which they arise. We describe how stability analyses of the vast parameter space relevant to TMT phasing can be accomplished in an efficient way and present examples of results. These concepts are also applicable to other multi-wavelength interferometric phase measurement techniques.

Effectiveness of null time-delay interferometry channels as instrument noise monitors in LISA

We present a study of the use and limits of the time-delay interferometry null channels for in-flight estimation of the Laser Interferometer Space Antenna instrumental noise. The paper considers how the two main limiting noise sources, test-mass acceleration noise and interferometric phase measurement noise, propagate through different time-delay interferometry channels: the Michelson combination $X$ that is the most sensitive to gravitational waves, then the less-sensitive combinations $\ensuremath{\alpha}$, and finally the null channel $\ensuremath{\zeta}$. We note that the null channel $\ensuremath{\zeta}$, which is known to be equivalent to any null channel, not only has a reduced sensitivity to the gravitational waves, but also features a larger degree of cancellation of the test mass acceleration noise relative to the interferometry noise. This severely limits its use in quantifying the low-frequency instrumental noise in the Michelson $X$ combination, which is expected to be dominated by acceleration noise. However, we show that one can still use in-flight noise estimations from $\ensuremath{\zeta}$ to put an upper bound on the considered noises entering in the $X$ channel, which allows one to distinguish them from a strong stochastic gravitational wave background.

Accurately measuring phase profiles of structured light in holographic optical tweezers

Accurate phase measurement of structured light is highly desirable in holographic optical tweezers, since quality of phase profiles have a strong impact on performance of manipulating particles. Here, we report a novel interferometric phase measurement technique for accurately measuring phase profile of structured light. The related experiments are performed, in which the measured phase profiles agree well with the designed phase profiles. It reveals that the phase profiles of structured light can be accurately obtained. Importantly, the fast switch between phase measurement and particle manipulation in the experiment can be easily implemented, which are strongly preferable in holographic optical tweezers. This study offers a promising way for accurately obtaining phase profiles of structure light, which is very helpful for particle manipulation.

Retrieve Ice Velocities and Invert Spatial Rigidity of the Larsen C Ice Shelf Based on Sentinel-1 Interferometric Data

The Larsen C Ice Shelf (LCIS) is the largest ice shelf in the Antarctica Peninsula, and its state can be considered to be an indicator of local climate change. The goal of this paper is to invert the rigidity of the LCIS based on the interferometric synthetic aperture radar (InSAR) technique using Sentinel-1 images. A targeted processing chain is first used to obtain reliable interferometric phase measurements under the circumstance of rapid ice flow. Unfortunately, only the descending data are available, which disallows the corresponding 2-D velocity field to be directly obtained from such measurements. A new approach is thus proposed to estimate the interferometric phase-based 2-D velocity field with the assistance of speckle tracking offsets. This approach establishes an implicit relationship between range and azimuth displacements based on speckle tracking observations. By taking advantage of such a relationship, the equivalent interferometric signals in the azimuth direction are estimated, thereby recovering the interferometric phase-based 2-D ice velocity field of the LCIS. To further investigate the state of the LCIS, the recovered 2-D velocity field is utilized to invert the ice rigidity. The shallow-shelf approximation (SSA) is the core of the reverse model, which is closely dependent on boundary conditions, including kinematic and dynamic conditions. The experimental results demonstrate that the spatial distribution of the rigidity varies approximately from 70 MPa·s1/3 to 300 MPa·s1/3. This rigidity distribution can reproduce a similar ice flow pattern to the observations.

Coastal Altimetry Using Interferometric Phase From GEO Satellite in Quasi-Zenith Satellite System

Global navigation satellite system reflectometry (GNSS-R) altimetry has great potential to provide high spatial–temporal resolution sea surface heights (SSHs) at low cost. Interferometric phase measurements between direct and reflected signals can be used for altimetry retrieval to achieve high-precision solutions. The motions of the medium Earth orbit (MEO) satellites cause interferometric phase change rapidly, which would increase the probability of occurrence of the phase unwrapping errors than the case of the geosynchronous Earth orbit (GEO) satellite. In order to overcome this problem, we propose a coastal GNSS-R altimetry algorithm using the signals from Quasi-Zenith Satellite System (QZSS) GEO satellite. Precise SSH variations can be achieved using the interferometric phase measurements without ambiguity fixed. We also perform coastal experiments on a trestle using a specialized GNSS-R setup to verify our algorithm. It is composed of an intermediate frequency (IF) data collector and two antennas. The up-looking antenna is used to receive direct signals, while the down-looking antenna receives the signals reflected from the sea surface. Raw IF data sampled at 62 MHz are collected and processed to derive interferometric carrier phase delay measurements using a self-developed software-defined receiver. Approximately 7 h of reflector heights are retrieved at 1-min intervals and the solutions are evaluated via comparison with measurements provided by a 26-GHz altimetry radar located near the GNSS-R setups. The results show that the root mean square error (RMSE) of sea level estimation is about 1.4 cm by using the QZSS GEO data.

Phase characterisation of metalenses

Metalenses have emerged as a new optical element or system in recent years, showing superior performance and abundant applications. However, the phase distribution of a metalens has not been measured directly up to now, hindering further quantitative evaluation of its performance. We have developed an interferometric imaging phase measurement system to measure the phase distribution of a metalens by taking only one photo of the interference pattern. Based on the measured phase distribution, we analyse the negative chromatic aberration effect of monochromatic metalenses and propose a feature size of metalenses. Different sensitivities of the phase response to wavelength between the Pancharatnam-Berry phase-based metalens and propagation phase-reliant metalens are directly observed in the experiment. Furthermore, through phase distribution analysis, it is found that the distance between the measured metalens and the brightest spot of focusing will deviate from the focal length when the metalens has a low nominal numerical aperture, even though the metalens is ideal without any fabrication error. We also use the measured phase distribution to quantitatively characterise the imaging performance of the metalens. Our phase measurement system will help not only designers optimise the designs of metalenses but also fabricants distinguish defects to improve the fabrication process, which will pave the way for metalenses in industrial applications.

Long cavity photonic crystal laser in FDML operation using an akinetic reflective filter.

A novel configuration of a Fourier domain mode locked (FDML) laser based on silicon photonics platform is presented in this work that exploits the narrowband reflection spectrum of a photonic crystal (PhC) cavity resonator. Configured as a linear Fabry-Perot laser, forward biasing of a p-n junction on the PhC cavity allowed for thermal tuning of the spectrum. The modulation frequency applied to the reflector equalled the inverse roundtrip time of the long cavity resulting in stable FDML operation over the swept wavelength range. An interferometric phase measurement measured the sweeping instantaneous frequency of the laser. The silicon photonics platform has potential for very compact implementation, and the electro-optic modulation method opens the possibility of modulation speeds far beyond those of mechanical filters.

Measurement of the Pancharatnam–Berry phase in two-beam interference

Young’s dual-pinhole interference experiment with arbitrary fully correlated and polarized vector light fields leads to a Pancharatnam–Berry geometric phase that is related to the associated dynamical phase. We demonstrate theoretically and experimentally how the dynamical phase across the interference pattern can be deciphered from the total phase, thereby leaving only the geometric phase. Our results constitute the first genuine interferometric phase measurements that yield the Pancharatnam–Berry geometric phase in Young’s two-beam interference setup.

Coherent beam combining of 61 femtosecond fiber amplifiers.

We report on the coherent beam combining of 61 femtosecond fiber chirped-pulse amplifiers in a tiled-aperture configuration along with an interferometric phase measurement technique. Relying on coherent beam recombination in the far field, this technique appears suitable for the combination of a large number of fiber amplifiers. The 61 output beams are stacked in a hexagonal arrangement and collimated through a high fill factor hexagonal micro-lens array. The residual phase error between two fibers is as low as λ/90 RMS, while a combining efficiency of ∼50% is achieved.

Microwave interferometer for phase and response time measurements.

We report the development of a microwave interferometer using a quadrature intermediate frequency (IQ) mixer designed to measure the relative phase change and response time of microwave devices tested in the Ka and upper K bands (22-40 GHz). The interferometer is currently used to test liquid crystal based devices. The system allows for the application of an AC bias beyond the amplitude/frequency limitations imposed on vector network analyzer bias ports. Our IQ mixer based design uses bias signals ranging from 0 to 100 V peak-to-peak in a frequency range from DC to 100 kHz. This range of bias signals is necessary to properly test the response of microwave devices designed with liquid crystal materials. The setup enables us to measure changes in the output phase of the device as a function of both the voltage and frequency of the applied bias signal. The setup also measures the phase difference as a function of microwave frequency and response times for the device under test. Our system can be integrated into a stand-alone test setup without the need for a vector network analyzer.

On Determining Surface Elevation Using a Synthetic-Aperture Radar in the Monostatic Noncoherent Mode

The article discusses the possibility of determining surface elevation using a conventional spaceborne synthetic aperture radar (SAR) survey by one satellite without interferometric phase measurements of the slant range. The capabilities of such determination are examined. A schematic diagram is given. Advantages and disadvantages are analyzed.

Single-Shot Phase and Amplitude Fluctuations of Narrow-Line Pulse Bursts in Divided-Pulse Amplifier

We demonstrate single-shot temporal-based interferometric phase measurements for bursts of 35 pulses of 1-ns duration in an Yb-doped fiber amplifier chain seeded by a 1063-nm single-frequency diode laser and operating in a highly nonlinear and saturated regime. We determined a maximum intra-pulse phase distortion ( $B$ -integral) of 21.2 rad by mixing the burst with a reference. The difference in the B-integral between the first and last pulse in the burst was large, 12.8 rad, but can be compensated for. Moreover, burst-to-burst amplitude fluctuations (standard deviation, std) were below 6%, where phase fluctuations were ~0.6 rad (std) or less. Even if these fluctuations cannot be compensated for, they still allow for efficient coherent combination, which opens up for transform-limited or narrow-line high-energy divided-pulse amplification with single-frequency seeding.

A Free-Space Interferometer for Phase-Delay Measurements in Integrated Optical Devices in Degenerate Pump-and-Probe Experiments

The authors report on a free-space interferometer for simultaneous phase and transmittance measurements of an optical signal in an integrated photonic circuit during degenerate pump-and-probe experiments. Differentiation of the weak probe signal from the strong pump relies on a lock-in amplifier. The interferometer shows high flexibility in terms of operating wavelengths, tested devices, and optical powers; it works with any integrated optical device provided with optical input–output channels; it can operate both with high input power (up to 100 mW at the sample input port) and low output power ( $ at the detection stage). Measuring both the transmittance and the phase of the optical signal allows using the phasor representation in the analysis. A dynamic operating mode allows to perform power and/or wavelength scan of the input signal, obtaining transmittance and phase spectra of the tested device. Remote control of the system and insulation from the external environment brings to more stable operating conditions. Characterization of the system properties with photonic-integrated structures (waveguide and ring resonator) is reported, with observed phase stability as high as 0.25°/min and average phase noise below 1°.

Coherent beam combining of seven fiber chirped-pulse amplifiers using an interferometric phase measurement.

Coherent beam combining in tiled-aperture configuration is demonstrated on seven femtosecond fiber amplifiers using an interferometric phase measurement technique. The residual phase error between two fibers is as low as λ/55 RMS and a combination efficiency of 48% has been achieved. The combined pulses are compressed to 216 fs, delivering 71 W average power at a repetition rate of 55 MHz. Operating the laser system in a nonlinear regime with an estimated B-integral of 5 rad yields a combining efficiency of 45% with the same phase stability. These results pave the way to very large high-power and high energy coherent beam combining systems.

Unconditional violation of the shot-noise limit in photonic quantum metrology

Interferometric phase measurement is widely used to precisely determine quantities such as length, speed, and material properties. Without quantum correlations, the best phase sensitivity $\Delta\varphi$ achievable using $n$ photons is the shot noise limit (SNL), $\Delta\varphi=1/\sqrt{n}$. Quantum-enhanced metrology promises better sensitivity, but despite theoretical proposals stretching back decades, no measurement using photonic (i.e. definite photon number) quantum states has truly surpassed the SNL. Rather, all such demonstrations --- by discounting photon loss, detector inefficiency, or other imperfections --- have considered only a subset of the photons used. Here, we use an ultra-high efficiency photon source and detectors to perform unconditional entanglement-enhanced photonic interferometry. Sampling a birefringent phase shift, we demonstrate precision beyond the SNL without artificially correcting our results for loss and imperfections. Our results enable quantum-enhanced phase measurements at low photon flux and open the door to the next generation of optical quantum metrology advances.

Highly scalable femtosecond coherent beam combining demonstrated with 19 fibers.

Coherent beam combining in the femtosecond regime of a record number of 19 fibers is demonstrated. The interferometric phase measurement technique, particularly well suited to phase-lock a very large number of fibers, is successfully demonstrated in the femtosecond regime. A servo loop is implemented to control piezoelectric fiber stretchers for both phase and delay variation compensation. The residual phase errors are below λ/60 rms. Nearly 50% of the total energy is contained in the far-field central lobe. After compression, we obtain a combined pulse width of 300 fs identical to the master oscillator pulse width.

Highly compact fiber Fabry-Perot interferometer: A new instrument design.

This paper presents the design, construction, and characterization of a new optical-fiber-based, low-finesse Fabry-Perot interferometer with a simple cavity formed by two reflecting surfaces (the end of a cleaved optical fiber and a plane, reflecting counter-surface), for the continuous measurement of displacements of several nanometers to several tens of millimeters. No beam collimation or focusing optics are required, resulting in a displacement sensor that is extremely compact (optical fiber diameter 125 μm), is surprisingly tolerant of misalignment (more than 5°), and can be used over a very wide range of temperatures and environmental conditions, including ultra-high-vacuum. The displacement measurement is derived from interferometric phase measurements using an infrared laser source whose wavelength is modulated sinusoidally at a frequency f. The phase signal is in turn derived from changes in the amplitudes of demodulated signals, at both the modulation frequency, f, and its harmonic at 2f, coming from a photodetector that is monitoring light intensity reflected back from the cavity as the cavity length changes. Simple quadrature detection results in phase errors corresponding to displacement errors of up to 25 nm, but by using compensation algorithms discussed in this paper, these inherent non-linearities can be reduced to below 3 nm. In addition, wavelength sweep capability enables measurement of the absolute surface separation. This experimental design creates a unique set of displacement measuring capabilities not previously combined in a single interferometer.

Novel homodyne frequency-shifting interference pattern locking system

We present a novel homodyne frequency-shifting interference pattern locking system to enhance the exposure contrast of interference lithography and scanning beam interference lithography (SBIL). The novel interference pattern locking system employs a special homodyne redundant phase measurement interferometer (HRPMI) as the sensor and an acousto-opto modulator (AOM) as the actuator. The HRPMI offers the highly accurate value as well as the direction recognition of the interference pattern drift from four quadrature interference signals. The AOM provides a very fine resolution with a high speed for phase modulation. A compact and concise system with a short optical path can be achieved with this new scheme and a small power laser head in tens of microwatts is sufficient for exposure and phase locking, which results in a relatively low-cost system compared with the heterodyne system. More importantly, the accuracy of the system is at a high level as well as having robustness to environmental fluctuation. The experiment results show that the short-time (4 s) accuracy of the system is ±0.0481 rad(3σ) at present. Moreover, the phase of the interference pattern can also be set arbitrarily to any value with a high accuracy in a relatively large range, which indicates that the system can also be extended to the SBIL application.

Time series analysis of InSAR data: Methods and trends

Time series analysis of InSAR data has emerged as an important tool for monitoring and measuring the displacement of the Earth’s surface. Changes in the Earth’s surface can result from a wide range of phenomena such as earthquakes, volcanoes, landslides, variations in ground water levels, and changes in wetland water levels. Time series analysis is applied to interferometric phase measurements, which wrap around when the observed motion is larger than one-half of the radar wavelength. Thus, the spatio-temporal “unwrapping” of phase observations is necessary to obtain physically meaningful results. Several different algorithms have been developed for time series analysis of InSAR data to solve for this ambiguity. These algorithms may employ different models for time series analysis, but they all generate a first-order deformation rate, which can be compared to each other. However, there is no single algorithm that can provide optimal results in all cases. Since time series analyses of InSAR data are used in a variety of applications with different characteristics, each algorithm possesses inherently unique strengths and weaknesses. In this review article, following a brief overview of InSAR technology, we discuss several algorithms developed for time series analysis of InSAR data using an example set of results for measuring subsidence rates in Mexico City.