About the application of the van Cittert-Zernike theorem inultrasonic imaging

The specific circumstances under which the van Cittert-Zernike theorem applies in ultrasonic imaging systems are examined through analysis and computations. Expressions are obtained for the mutual coherence function of an incoherent source when the signals are discrete in time and space and have finite lengths. Expressions are also obtained for statistics and effective signal-to-noise ratios that describe the error in the assumption of an incoherent source with finite signal lengths. Images of a one-dimensional source are reconstructed for different signal lengths and different pulse windows. The results show that ultrasonic signals with a relatively long effective length are needed to satisfy the incoherence requirement for image reconstruction based on the van Cittert-Zernike theorem. Consequently, although the van Cittert-Zernike theorem may be used to estimate the coherence length of ultrasonic signals in the aperture of an imaging system, special data acquisition techniques are needed for satisfactory reconstruction of ultrasonic images when depth resolution like that in current b-scans is required     

The Spatial Coherence Properties of Gaussian Schell-model Beams

The transverse and longitudinal spatial coherence properties of the light beams generated by planar gaussian Schell-model sources are discussed. It is found that for all gaussian Schell-model beams the ratio of the transverse coherence length to the beam width remains invariant upon propagation. An examination of the longitudinal coherence for both on-axis and off-axis pairs of points indicates that the longitudinal coherence will not, in general, die out as the separation between the points is increased. Rather, the degree of longitudinal coherence will approach a finite (non-zero) value as long as the source contains a finite coherence area, regardless of how small this area may be. Gaussian quasihomogeneous beams are studied as a limiting case. The relation of the present work to the analysis of speckle size is briefly discussed.     

Coherence analysis of optical frequency-modulated continuous-wave interference

I analyze the coherence of optical frequency-modulated continuous-wave (FMCW) interference. With a simple model modified from the classical coherence theory, I successfully derive the relationships among the frequency bandwidth, coherence length, and coherence time of the practical optical source, and the contrast of the beat signal in optical FMCW interference.     

Superfluid3He in restricted geometries

It is shown that finite size effects stabilize a variety of phases in superfluid³He which do not occur in the bulk liquid. These phases are formed at temperatures at which the coherence length is comparable with the smallest linear dimension of the system. Right circular cylindrical geometries are considered explicitly.     

A Multiple-beam Interferometer Coherence Analyser

A laser coherence analyser is described which yields simultaneous temporal and spatial coherence analysis of laser beams. It is of particular application to high power pulsed laser systems, for high speed holography and non-linear optics, where the short duration of the pulse (often less than 50 nsec) excludes the use of photoelectric spectrometers, employing time variation scanning of the optical path length. The instrument consists of two high quality spherical mirrors, having dielectric multilayer coatings of reflectivity typically 0.95 to 0.97, separated by a cylindrical invar spacer of adjustable length. In use, the spacer is normally set so that the pole separation of the plates is a few hundred micrometres less than their common radius of curvature (typically 10 cm). The resulting defocusing term combines with spherical aberration to yield a multiple beam interferogram having an annular region of nearly linear dispersion. Spectral resolving powers in the range 10⁷ to 10⁸ are readily obtainable. When the input beam possesses a high degree of spatial coherence, the fringes of the concentric rings interference pattern are alternately of high and low intensity, the visibility of the alternation being a direct measure of the spatial coherence. This alternation effect may be eliminated by obstructing one-half of the input aperture. Any spatial, spectral or intensity variations across the laser beam are superposed on the fringe pattern, since the spherical interferometer is not translationally invariant in a parallel beam. This is in contrast to the behaviour of a plane Fabry-Pérot etalon. The experimental characteristics of the instrument, and its properties, as predicted by numerical computation, are examined both for C.W. and pulsed lasers. The effect of the finite response time is described. This is particularly important when the coherence analyser is employed in conjunction with an image tube streak camera to obtain time-resolved spectra of pulsed systems. Both time-integrated and time-resolved fringe profiles are considered for various types of laser pulse. The effect of plate defects on fringe profiles is also considered in outline. In addition to providing quasi-linear dispersion and information on coherence and other spatial properties of the input, the instrument described has the advantages of providing high illumination, and of being easy to use in practice, the plates remaining permanently in alignment.     

Estimation of longitudinal resolution in optical coherence imaging

The spectral shape of a source is of prime importance in optical coherence imaging because it determines several aspects of image quality, especially longitudinal resolution. Wide spectral bandwidth, which provides short coherence length, is sought to obtain high-resolution imaging. To estimate longitudinal resolution, the spectral shape of a source is usually assumed to be Gaussian, although the spectra of real sources are typically non-Gaussian. We discuss the limit of this assumption regarding the estimation of longitudinal resolution. To this end, we also investigate how coherence length is related to longitudinal resolution through the evaluation of different definitions of the coherence length. To demonstrate our purpose, the coherence length for several theoretical and real spectral shapes of sources having the same spectral bandwidth and central wavelength is computed. The reliability of coherence length computations toward the estimation of longitudinal resolution is discussed.     

Evaluation of the impact of finite-resolution effects on scintillation compensation using two deformable mirrors.

The impact of finite-resolution deformable mirrors and wave-front sensors is evaluated as it applies to fullwave conjugation using two deformable mirrors. The first deformable mirror is fixed conjugate to the pupil, while the second deformable mirror is at a finite range. The control algorithm to determine the mirror commands for the two deformable mirrors is based on a modification of the sequential generalized projection algorithm. The modification of the algorithm allows the incorporation of Gaussian spatial filters into the optimization process to limit the spatial-frequency content applied to the two deformable mirrors. Simulation results are presented for imaging and energy projection scenarios that establish that the optimal spatial filter waist to be applied is equal to the subaperture side length in strong turbulence. The effect of varying the subaperture side length is examined, and it is found that to effect a significant degree of scintillation compensation, the subapertures, and corresponding spacing between actuators, must be much smaller than the coherence length of the input field.     

Determination of the Coherence Length and the Cooper-Pair Size in Unconventional Superconductors by Tunneling Spectroscopy

The main purpose of the paper is to discuss a possibility of the determination of the values of the coherence length and the Cooper-pair size in unconventional superconductors by using tunneling spectroscopy. In the mixed state of type-II superconductors, an applied magnetic field penetrates the superconductor in the form of vortices which form a regular lattice. In unconventional superconductors, the inner structure of a vortex core has a complex structure which is determined by the order parameter of the superconducting state and by the pairing wavefunction of the Cooper pairs. In clean superconductors, the spatial variations of the order parameter and the pairing wavefunction occur over the distances of the order of the coherence length and the Cooper-pair size, respectively. Therefore, by performing tunneling spectroscopy along a line passing through a vortex core, one is able, in principle, to estimate the values of the coherent length and the Cooper-pair size. To our knowledge, the theoretical consideration of the density of states inside vortex cores for unconventional superconductors is presented for the first time.     

Experimental evidence of phase coherence of magnetohydrodynamic turbulence in the solar wind: GEOTAIL satellite data

Magnetohydrodynamic (MHD) turbulence is commonly observed in the solar wind. Nonlinear interactions among MHD waves are likely to produce finite correlation of the wave phases. For discussions of various transport processes of energetic particles, it is fundamentally important to determine whether the wave phases are randomly distributed (as assumed in the quasi-linear theory) or have a finite coherence. Using a method based on the surrogate data technique, we analysed the GEOTAIL magnetic field data to evaluate the phase coherence in MHD turbulence in the Earth's foreshock region. The results demonstrate the existence of finite phase correlation, indicating that nonlinear wave-wave interactions are in progress.     

Cross-talk analysis of time-division multiplexing of polarization-insensitive fiber-optic Michelson interferometric sensors with a 3 x 3 directional coupler

A time division multiplexing of polarization-insensitive fiber-optic Michelson interferometric sensors (TDM-PIMI's) with a 3 x 3 directional coupler is presented. The elimination of polarization-induced fading and the output intensities of the TDM-PIMI system are described and demonstrated. The output intensity of each sensor of the system can be demodulated by a passive homodyne method to increase the sensor bandwidth significantly. The sensor cross talk of the system having an optical gate with a finite extinction ratio is analyzed. The use of a laser source with an adequate coherence length to reduce the sensor cross talk is suggested. The delay-fiber cross talk of the system by Rayleigh backscattering is analyzed and demonstrated. We further suggest some methods that could possibly reduce the effect of the Rayleigh backscattered light. Finally a sophisticated design of a TDM-PIMI system with a 3 x 3 directional coupler is described.     

Radiometric theory of spatial coherence in free-space propagation

The radiometric theory of spatial coherence is presented with special attention to the validity of the approximations on which it is based. A new definition of the transverse coherence area is introduced and shown to be in general agreement with earlier definitions. In free-space propagation the product of the transverse coherence area and the intensity is shown to be constant along rectilinear rays, and, for radiation from uniform Lambert sources, a well-known paraxial formula for the transverse coherence area is extended to the extraparaxial domain. A decrease of the spatial coherence in free-space propagation takes place in regions with an increase of the intensity. For imaging systems this occurs in a finite part of image space whenever a real image of a diffusely radiating, extended object is formed at a finite distance.     

Detection of coupling between oscillators from their short time series: Condition of applicability of the method of phase dynamics modeling

One of the most sensitive methods for determining the presence and character of coupling between two oscillators is based on the modeling of their phase dynamics. This method is applicable if the observed oscillations are not synchronous and the length of available time series is no less than 50 characteristic periods. Now it is established that the method can be applied to still shorter time series with a length of 20 to 50 periods, provided that the estimated phase coherence coefficient (measuring the degree of synchronization) does not exceed a certain threshold dependent on the time series length and the frequency detuning. This result expands the possibilities of detecting the coupling between oscillatory systems under conditions of nonstationary signals and deficient data.     

Crossover Effects in the Nonlinear σ-Model with Damping

We study the behavior of the nonzero temperature quantum nonlinear sigma model in d = 2 dimensions at finite temperatures in the presence of the damping term. Using the renormalization group (RG) method we calculated the effect of the finite temperature on the magnetic coherence length and the damping of the fluctuations. The results have been used to calculate the relaxation rates from nuclear magnetic resonance, which have been analyzed phenomenologically by different authors in order to get the phase diagram in the doped cuprate superconductors. Our calculations showed a good agreement with the experimental data, but we also evaluated the corrections which appear due to a more accurate analysis performed using the RG method.     

Performance comparison of an all-fiber-based laser Doppler vibrometer for remote acoustical signal detection using short and long coherence length lasers

All-fiber laser Doppler vibrometer systems have great potential in the application of remote acoustic detection. However, due to the requirement for a long operating distance, a long coherence length laser is required, which can drive the system cost high. In this paper, a system using a short coherence length laser is proposed and demonstrated. Experimental analysis indicates that the multi-longitudinal modes of the laser cause detection noise and that the unequal length between two paths (local oscillator path and transmission path) increases the intensity and the frequency components of the noise. In order to reduce the noise, the optical length of the two paths needs to be balanced, within the coherence length of the source. We demonstrate that adopting a tunable optical delay to compensate the unequal length significantly reduces the noise. In a comparison of the detection results by using a short coherence laser and a long coherence laser, our developed system gives a good performance on the acoustic signal detection from three meters away.     

Generation of entangled photon pairs using small-coherence-time continuous wave pump lasers

We address the generation of entangled photon pairs by parametric downconversion from solid state cw lasers with small coherence time. We consider a compact and low-cost setup based on a two-crystal scheme with type-I phase matching. We reconstruct the full density matrix by quantum tomography and analyze in detail the entanglement properties of the generated state as a function of the crystal's length and the coherence time of the pump. We verify the possibility to improve the visibility using a purification protocol based on a compensation crystal.     

Theory of partial coherence for weakly periodic media

The second-order theory of partial coherence for scalar and TE or TM fields is developed for weakly periodic media, and the van Cittert-Zernike theorem of classical coherence theory is generalized for such media. The coherence properties of a wave field, generated by a quasi-homogeneous source distribution at the entrance plane of a finite weakly periodic medium, are calculated both inside such a structure and in the far field. The second-order theory of partial coherence for pulse propagation through weakly periodic media is also developed.     

Fundamental characteristics of a synthesized light source for optical coherence tomography

We describe the fundamental characteristics of a synthesized light source (SLS) consisting of two low-coherence light sources to enhance the spatial resolution for optical coherence tomography (OCT). The axial resolution of OCT is given by half the coherence length of the light source. We fabricated a SLS with a coherence length of 2.3 microm and a side-lobe intensity of 45% with an intensity ratio of LED1:LED2 = 1:0.5 by combining two light sources, LED1, with a central wavelength of 691 nm and a spectral bandwidth of 99 nm, and LED2, with a central wavelength of 882 nm and a spectral bandwidth of 76 nm. The coherence length of 2.3 microm was 56% of the shorter coherence length in the two LEDs, which indicates that the axial resolution is 1.2 microm. The lateral resolution was measured at less than 4.4 microm by use of the phase-shift method and with a test pattern as a sample. The measured rough surfaces of a coin are illustrated and discussed.     

Dynamics of adaptive optical systems

Dynamic (time) characteristics of adaptive systems are analyzed. A common adaptive system with a finite frequency band (or a finite response time) is described as a dynamic constant time-delay system, where time delay is to be much shorter than the time of coherence radius transfer through an optical beam by a mean wind speed. The questions of coherent beam formation are considered with use of the reference source. The analytical calculation of the Strehl parameter is made on basis of the generalized Huygens-Kirchhoff principle. An adaptive system is considered where the correcting phase is calculated with the use of both its derivatives and the signal, as well as adaptive systems using different time-predicting algorithms of the correcting signal for future time points. The use of a predicted phase front of the correcting wave allows much longer time delays. The stronger the phase distortions in the optical wave, the higher the time gain in comparison with common (with constant time delay) adaptive systems.     

Multipass Michelson interferometer with the use of a wavelength-modulated laser diode

An improved multipass Michelson interferometer is implemented. This technique uses the fact that the wavelength of a laser diode varies in proportion to the diode's injection current. With this method the sensitivity augmentation is accomplished by inserting a beam splitter into one arm of the interferometer, resulting in multiple reflections between the end mirror and the beam splitter. In addition, the interference of laser beams reflected from two arms can be accomplished with unequal arms in the condition of a short coherence length. The sensitivity increase of interference fringes and the compensation of the short coherence length have been demonstrated in experiments.     

Effect of wavefront corrugations on fringe motion in an astronomical interferometer with spatial filters

Numerical simulations of atmospheric turbulence and adaptive optics (AO) wavefront correction are performed to investigate the time scale for fringe motion in optical interferometers with spatial filters. These simulations focus especially on partial AO correction, where only a finite number of Zernike modes are compensated. The fringe motion is found to depend strongly on both the aperture diameter and the level of AO correction used. In all the simulations the coherence time scale for interference fringes is found to decrease dramatically when the Strehl ratio provided by the AO correction is < or = 30%. For AO systems that give perfect compensation of a limited number of Zernike modes, the aperture size that gives the optimum signal for fringe phase tracking is calculated. For AO systems that provide noisy compensation of Zernike modes (but are perfectly piston neutral), the noise properties of the AO system determine the coherence time scale of the fringes when the Strehl ratio is < or = 30%.