Interferometric Phase Measurements Research Articles

Fibre interferometer for multi-wavelength interferometry with a femtosecond laser

In this paper we propose a novel fibre interferometer for the simultaneous measurement of optical phase at multiple wavelengths generated within the spectral envelope of a femtosecond laser. The use of narro wb andwidth fibre Bragg gratings allows particular wavelengths to be selected with sufficiently narrow bandwidth to give the necessary coherence length for interferometric phase measurements over practical distances exceeding 30 mm. Multi-wavelength interferometry provides a means of making absolute measurements by extending the unambiguous measurement range beyond the wavelength of light. Preliminary experimental results are presented for a two-wavelength sensor where an absolute range of 0.13 mm with 3 nm resolution has been achieved producing a dynamic range of 43 000:1. This approach has the potential to be applied to high speed transient events.

Prospecting for horizontal surface displacements in Antelope Valley, California, using satellite radar interferometry

We compare interferometric synthetic aperture radar displacement maps for Antelope Valley, California, derived for two different satellite orbit geometries to test the common assumption that surface displacements above deforming aquifer systems are purely vertical. This assumption greatly simplifies models of aquifer system deformation related to changes in groundwater flow and storage, permitting the use of one‐dimensional codes for characterizing the compaction occurring in the flow model. The analysis is complicated by interferometric phase noise and atmospheric signal contributions. We use a variance‐weighted least squares approach to optimally use the information contained in several interferograms. Our analysis of the displacement maps does not identify significant displacement differences for the two acquisition geometries, implying that the motions are indeed nearly vertical. Uncertainties in the interferometric phase measurement and atmospheric delay signals preclude any conclusion regarding the existence of horizontal displacements smaller than ∼2 cm. The absence of spatial correlation between the observable differences and the known subsidence field indicates that horizontal displacements are indeed negligible for inelastic aquifer system compaction in Antelope Valley.

A method for measuring the complex refractive index and thickness of a thin metal film

A circularly polarized heterodyne light beam is incident on a thin metal film, causing successive reflections and refractions to occur at the two sides of the thin film. The phase difference between p- and s-polarizations of the multiple-beam interference signal can be measured accurately with an analyzer and heterodyne interferometry. The phase difference depends on the azimuth angle of the analyzer, the complex refractive index and the thickness of the thin metal film. The measured values of the phase differences under three different azimuth angles of the analyzer can be substituted into the special equations derived from Fresnel’s equations and multiple-beam interference. Hence, the complex refractive index and the thickness of the thin metal film can be estimated by using a personal computer with a numerical analysis technique. Because of its common-path optical configuration and its heterodyne interferometric phase measurement, this method has many merits, such as high stability against surrounding vibrations, high resolution and easy operation.

Volumetric phase-measuring interferometer for three-dimensional coordinate metrology

We present a volumetric interferometer that has been specially designed to improve the measurement accuracy of the three-dimensional coordinate metrology by reducing excessive Abbe offsets. The interferometer is composed of two main parts; one is a three-dimensionally movable target and the other is a stationary two-dimensional array of photo-detectors. The target made of point diffraction sources emanates two spherical wavefronts, of which interference intensity is monitored by the photo-detectors. Applying phase shifting allows the phases of the photo-detectors to be precisely obtained, and the measured phases are fitted to a geometric model of multilateration so as to determine the xyz-location of the target by minimizing least square errors. A prototype built on a CMM demonstrates that the proposed interferometer is capable of measuring the xyz-coordinates of the probe with a volumetric uncertainty of less than 1.0 μm.

Advanced interferometric phase and amplitude noise measurements

The measurement of the close-to-the-carrier noise of two-port radio frequency and microwave devices is a relevant issue in time and frequency metrology and in some fields of electronics, physics, and optics. While phase noise is the main concern, amplitude noise is often of interest. Presently the highest sensitivity is achieved with the interferometric method, that consists of the amplification and synchronous detection of the noise sidebands after suppressing the carrier by vector subtraction of an equal signal. A substantial progress in understanding the flicker noise mechanism of the interferometer results in new schemes that improve by 20–30 dB the sensitivity at low Fourier frequencies. These schemes, based on two or three nested interferometers and vector detection of noise, also feature closed-loop carrier suppression control, simplified calibration, and intrinsically high immunity to mechanical vibrations. This article provides the complete theory and detailed design criteria, and reports on the implementation of a prototype working at the carrier frequency of 100 MHz. In real-time measurements, a background noise of −175 to −180 dBrad2/Hz has been obtained at f=1 Hz off the carrier; the white noise floor is limited by the thermal energy kBT0 referred to the carrier power P0 and by the noise figure of an amplifier. Exploiting correlation and averaging in similar conditions, the sensitivity exceeds −185 dBrad2/Hz at f=1 Hz; the white noise floor is limited by thermal uniformity rather than by the absolute temperature. A residual noise of −203 dBrad2/Hz at f=250 Hz off the carrier has been obtained, while the ultimate noise floor is still limited by the averaging capability of the correlator. This is equivalent to a signal-to-noise (S/N) ratio of 2×1020 with a frequency spacing of 2.5×10−6. All these results have been obtained in a relatively unclean electromagnetic environment, and without using a shielded chamber. Implementation and experiments at that sensitivity level require skill and tricks, for which a great effort is spent in the article. Applications include the measurement of the properties of materials and the observation of weak flicker-type physical phenomena, out of reach for other instruments. As an example, we measured the flicker noise of a by-step attenuator (−171 dBrad2/Hz at f=1 Hz) and of the ferrite noise of a reactive power divider (−173.7 dBrad2/Hz at f=1 Hz) without need of correlation. In addition, the real-time measurements can be exploited for the dynamical noise correction of ultrastable oscillators.

A Data Conversion Technique for Accurate Interferometric Testing of Spherical Surfaces.

This paper presents a Cartesian to polar conversion technique based on a polynomial interpolation. The method is used for the processing of data from the two-orientation method for measuring spherical surfaces with an interferometer. The interferometer output consists of points on a square-grid coordinate system, which are then converted to a polar-grid coordinate system. The profile height data of the points on the polar-grid are calculated from a polynomial neighborhood analysis of points on the square-grid. Since the converted point on the polar-grid does not need to be at the center of the points in the analysis, profiles along concentric circles near to the edge of the test surface can also be calculated. A simulation of the polynomial and conventional 4-point techniques is carried out to verify the process. The comparison shows that the polynomial technique has produce a higher data conversion accuracy. Experiments on a phase-measuring interferometer were also conducted to verify the approach.

Optimal input states and feedback for interferometric phase estimation

We derive optimal N-photon two-mode input states for interferometric phase measurements. Under canonical measurements the phase variance scales as N-2 for these states, as compared to N-1 or N-1/2 for states considered bq previous authors. We prove, that it is not possible to realize the canonical measurement by counting photons in the outputs of the interferometer, even if an adjustable auxiliary phase shift is allowed in the interferometer. However. we introduce a feedback algorithm based on Bayesian inference to control this auxiliary phase shift. This makes the measurement close to a canonical one, with a phase variance scaling slightly above N-2. With no feedback, the best result (given that the phase to be measured is completely unknown) is a scaling of N-1. For optimal input states having up to four photons, our feedback scheme is the best possible one, but for higher photon numbers more complicated schemes perform marginally better.

Interferometry-based measurements of oil-film thickness

Measurement of the thickness of thin lubricant films separating rotating surfaces in elastohydrodynamic experiments presents some challenging problems. The nature of the experimental apparatus inhibits the use of most commonly applied interferometric phase measurement methods. Also the absolute thickness of the separating film must be determined, as opposed to relative distances that would be sufficient in most other measurement scenarios where interferometry methods are used. In this paper, computer-based analysis of interferograms recorded using an elastohydrodynamic lubrication Fitzeu interferometer (a so-called ball-and-disc apparatus) is discussed, the main objective being to extract the absolute oil-film thickness. Intensity based methods (most importantly, calibration look-up procedures where colour parameters from recorded dynamic interferograms are compared with table values corresponding to known film thicknesses, but also a phase measurement approach based on multi-channel interferometry using trichromatic light) are described. A discussion regarding compensation for measurement errors due to the pressure dependence of the refractive index of the lubricant is also included.

Interferometric phase measurements for polarization eigenvectors in twisted nematic liquid crystal spatial light modulators

We show interferometric measurements of the phase delays for both eigenvectors in a twisted nematic liquid crystal spatial light modulator (LCSLM). These polarization eigenvectors are of interest for newer LCSLMs that are thinner and consequently have a lower birefringence compared with earlier devices. We use a new technique in which an electro-optic light modulator provides a time varying phase shift between the electric field components in the two arms of a Mach–Zehnder interferometer. The appropriate eigenvectors are generated and detected for a LCSLM located in one arm of the interferometer. We then use a lock-in amplifier to accurately measure the phase delay for the LCSLM as a function of applied voltage. Our experimental results show agreement with the theory.

Interferometric radar measurements of water level changes on the Amazon flood plain

Measurements of water levels in the main channels of rivers, upland tributaries and floodplain lakes are necessary for understanding flooding hazards, methane production, sediment transport and nutrient exchange. But most remote river basins have only a few gauging stations and these tend to be restricted to large river channels. Although radar remote sensing techniques using interferometric phase measurements have the potential to greatly improve spatial sampling, the phase is temporally incoherent over open water and has therefore not been used to determine water levels. Here we use interferometric synthetic aperture radar (SAR) data, acquired over the central Amazon by the Space Shuttle imaging radar mission, to measure subtle water level changes in an area of flooded vegetation on the Amazon flood plain. The technique makes use of the fact that flooded forests and floodplain lakes with emergent shrubs permit radar double-bounce returns from water and vegetation surfaces, thus allowing coherence to be maintained. Our interferometric phase observations show decreases in water levels of 7-11 cm per day for tributaries and lakes within approximately 20 km of a main channel and 2-5 cm per day at distances of approximately 80 km. Proximal floodplain observations are in close agreement with main-channel gauge records, indicating a rapid response of the flood plain to decreases in river stage. With additional data from future satellite missions, the technique described here should provide direct observations important for understanding flood dynamics and hydrologic exchange between rivers and flood plains.

Wavelength-Tunable Laser-Diode Interferometer

Laser diodes (LDs) have been applied to a phase-measuring interferometer through the wavelength tunability of LDs by controlling their currents. Laser-diode interferometers based on a heterodyne technique are reviewed. A two-wavelength laser-diode interferometer is demonstrated with current control of dual LDs in opposite directions. A synthetic wavelength makes it possible to extend the range of interferometric measurements. The wavelength is controlled by the laser injection current and is stepwise or rampwise changed to introduce a time-varying phase difference between the two beams of an interferometer with unbalanced optical path lengths. The optical output is demodulated with a phase-extraction algorithm. Systematic phase errors caused by the LD-power variation and by the difference between the beat frequency and ramp frequency are analyzed. A feedback interferometer with electronics is used to eliminate the phase error by locking the interferometer on a preset phase. Typical experimental results are shown.

Very high frequency and microwave interferometric phase and amplitude noise measurements

The interferometric technique allows close-to-the-carrier measurements of both phase and amplitude noise, improving the instrument noise floor by 10–25 dB as compared to the traditional method based on a saturated mixer. Principles and basic equations describing the noise measurement system are given, together with design strategies suitable to microwave and very high frequency bands. Two prototypes, operating at 9 GHz and 100 MHz are discussed in detail. The relevant features of these prototypes are the capability to operate in a wide power range, below 0 dBm and above 20 dBm, and low noise floor. The latter is about −180 dB rad2/Hz (white) and 150 dB rad2/Hz (flicker) at 1 Hz Fourier frequency, at carrier power from 9 to 15 dBm.

Speckle intensity and phase gradients: influence on fringe quality in spatial phase shifting ESPI-systems

Spatial phase shifting (SPS) is a good method to exclude time-dependent phase variations from interferometric phase measurements, since the phase shifted data are acquired simultaneously but from different points. This leads to problems when SPS is applied to electronic speckle pattern interferometry. The speckled object wave contains a multitude of phase and intensity gradients that violate the assumption of constant object intensity and phase on adjacent data points, from which most phase reconstruction formulas are derived. Based on today's knowledge of second-order properties of speckle fields, some phase retrieval algorithms specially adapted to SPS are proposed that improve the phase maps obtained. Quantitative experimental results are given to estimate the amount of improvement.

High Power Interferometric Phase Measurement Limited by Quantum Noise and Application to Detection of Gravitational Waves

Interferometric detection of gravitational waves at levels of astrophysical interest is expected to require measurement of optical phase differences of $\ensuremath{\le}{10}^{\ensuremath{-}10}$ rad. To achieve this sensitivity the measurement must be made at the quantum (shot noise) limit. We have built a laboratory-scale interferometer to investigate limits to sensing the phase at this level, and obtained, with 70 W of circulating power, a phase sensitivity limited by shot noise of $3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}\phantom{\rule{0ex}{0ex}}\mathrm{rad}/\sqrt{\mathrm{Hz}}$ at frequencies above 1.5 kHz. We describe the experiment and discuss the implications for the Laser Interferometer Gravitational-Wave Observatory (LIGO).

SAR interferometry over Baltic Sea ice

SAR interferometry (InSAR) offers new interesting possibilities for research in sea ice radar scattering and sea ice mechanics. A case study of this is presented from the Baltic Sea in late March 1992. Interferometric coherence is mainly dependent of the temporal characteristics of the scattering sources in sea ice. Different areas with different scattering properties were examined and the present data indicates that more field data is necessary to fully understand the InSAR coherence over sea ice. However, some interesting features were noted. Over low-salinity ice, backscattering and coherence seems to be related, high backscatter areas are more unstable than low backscatter areas. Over areas with surface roughness scattering, the scattering is relatively stable and also that a snow cover seems to retain the coherence over such areas. Interferometric phase measurements are dependent on small deformations of the ice pack. Fast ice which is (nearly) stationary experiences small discontinuous slips and deformations. Interferometric phase measurements are very sensitive to these slips, displacements and deformations and will provide new insight into the rheology for fast ice and how the fast ice starts to move. How the fast ice starts to move is one of the major problems in sea ice mechanics research and there is not much earlier data on the subject. In the present case, the ice was nearly stationary as the stresses were below the yield limit under the low forcing conditions. Two ice floe compressions have been observed and the strains are believed to be viscous with a viscosity value at approximately 1013-1014kg (ms). Both the interferometric phase and the coherence measurements over ice are believed to be of great value in future backscattering models and sea ice mechanics models.

Phase properties of the optical near field

This paper presents a theoretical and numerical study of the phase properties of the optical near field. A model based on the first Rytov approximation for three-dimensional electromagnetic vector fields describes the relationship between the phase variations and both the topographic and optical properties of the scatterer. It is shown that strong polarization effects can lead to subwavelength phase variations around nanometric structures. The conclusions of the model are illustrated by exact numerical calculations. This study should find broad experimental applications in near-field optical interferometric phase measurements.

Derivation and error analysis of echo phase parameters for the dynasonde

Ionospherically reflected echoes are received with a four‐element multiplexed interferometer array and two phase‐matched receivers in implementations of the National Oceanic and Atmospheric Administration dynasonde. From these data, six phase parameters (Φ0, Φx, Φy, Φt, Φp, and Φƒ) are obtained and used to derive echo location, Doppler velocity, wave polarization, and virtual range. Since 2 π aliasing is an inherent feature of interferometric spaced antenna phase measurements, the phase parameters cannot be derived directly from the measured phase values using the method of least squares. In this work, we introduce a general procedure for the derivation of these parameters that (1) employs a “zero‐freedom” technique to derive initial estimates of the phase parameters, (2) derives shifted values of the measured phases from the six estimates, and (3) uses the method of least squares in conjunction with the shifted phases to improve the phase parameter estimates. This procedure minimizes the phase ambiguity inherent in interferometric phase measurements and derives phase parameters that approach the ideal least squares result. Furthermore, the ionospheric echo “quality” is quantified by two error parameters, defined asŝ, the least squares of the measured phase errors, and EP, the RMS phase error. It is also shown that the value of ŝ relative to the square of the standard deviation of measured phase is equal to the number of degrees of freedom in the phase parameters. Applying these techniques to data acquired with the Utah State University dynasonde, we derive a standard deviation of <2° in the measured echo phase.

A method for the spatial calibration of a commercial phase measuring interferometer — theoretical aspects

In a phase measuring interferometer (PMI) the interference pattern is focused on a charge coupled device (CCD) detector array and data points are sampled at the corresponding locations. In most commercially available systems, a zoom lens forms part of the optical configuration to suitably magnify the test optic to fill the detector. Thus, the spatial sampling conditions are variable and determined by the magnification of the optical system and the size and spatial locations of the elements of the CCD. For accurate assessment of the spatial properties of the test optic (including power spectra for example), an accurate determination of the sampling conditions must be made. A computer simulation has been developed that enables an investigation of the theoretical capabilities of sub-pixel calibration resolution. In this paper we demonstrate how sub-pixel measurements may be achieved and indicate some of the factors that influence the outcome of the measurement.

Evaluation of residual deformations generated by a pulsed ion implanter using interferometric phase measurement

A Fourier transform method of holographic fringe pattern analysis is applied to measure surface residual deformations generated by a pulsed ion implanter. The technique uses a fixture that makes it possible to remove the specimen and put it back into the same position after being implanted. The phase information from interferograms extracted by means of the Fourier transform method is unwrapped using an algorithm based on cellular automata. Results computed from the application of a numerical model are compared with those determined experimentally and a reasonable agreement is obtained.

Faraday-effect magnetometry: compensation for the temperature-dependent Verdet constant

A technique for simultaneous independent measurement of temperature and magnetic field in a Faraday-effect magnetometer is presented. The technique is based on interferometric phase measurement of temperature combined with magnetic-field-induced visibility modulation of the interferometric fringes. The technique is demonstrated using a paramagnetic FR-5 glass sensing element in a Michelson and a low-finesse Fabry-Perot configuration. The optical configuration is shown to be capable of remote operation using fibre-optic links. The measurement of temperature is shown to enable compensation of the scale factor for the temperature-dependent Verdet constant. Unlike in all previously reported techniques, the temperature is measured over exactly the same optical path as that subject to the magnetic field.