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Showing papers on "Phase (waves) published in 1970"


Journal ArticleDOI
TL;DR: In this paper, the Fourier transform of a function g(x) is quantized, and the function recovered by inverse transformation differs from g (x) by means of a biased limiter model.
Abstract: If the Fourier transform of a function g(x) is quantized, the function recovered by inverse transformation differs from g(x). By means of a biased limiter model, the effects of Fourier-domain phase...

174 citations


Journal ArticleDOI
E. Muehldorf1
TL;DR: In this paper, the phase center for an arbitrary plane is calculated from the E- and H -plane phase centers, and the dependence of the phase centers on horn dimensions is shown. But this is based on a vector approach, by deriving the phase centre from the expressions for the far field.
Abstract: The calculation of phase centers for rectangular and diagonal horns is presented. The calculation is based on a vector approach, by deriving the phase center from the expressions for the far field. Different expressions are derived for the phase center of the E and H planes. The phase center for an arbitrary plane is calculated from the E- and H -plane phase centers. Graphs are presented showing the dependence of the phase centers on horn dimensions.

106 citations


Journal ArticleDOI
TL;DR: An attempt is made to assess the relative saliency of the power and phase variables by adding the phase cue as a distracting feature to the power‐discrimination task.
Abstract: Pairs of very brief pulses can be arranged to produce transient signals having spectra that differ only in power, only in phase, or in both power and phase. Conventional intensity discrimination for such click pairs is obtained by using the signals that differ only in their power spectra. Some of the corresponding signals that differ only in their phase spectra are also readily discriminable. An attempt is made to assess the relative saliency of the power and phase variables by adding the phase cue as a distracting feature to the power‐discrimination task. The presence of such an uncorrelated phase cue can lower the detectability of power differences.

69 citations


Proceedings ArticleDOI
11 May 1970
TL;DR: In this paper, the principle of operation, physical realization, and performance parameters for a reciprocal phase shifter of this latter type were described, as well as the operation of a dual-mode transmission line in which the two nonreciprocal phase shifters occupied the same physical space.
Abstract: An attractive method for producing reciprocal phase shift is to use a pair of identical nonreciprocal phase shifters with circulators at input and output, as shown in Figure 1. In this scheme, signals passing from left to right are sent through the lower phase shifter, while signals passing from right to left are sent through the upper phase shifter. By switching the phase shifters in a complementary manner, equal variable insertion phases can be provided for either direction of propagation. This approach to achieving reciprocal phase shift yields the desirable bandwidth, figure of merit, and temperature stability advantages of nonreciprocal phase shifters, except that it is more complicated and has the additional losses of the input and output circulators. However, the added complexity can be significantly reduced by employing a dual-mode transmission line in which the two nonreciprocal phase shifters occupy the same physical space. It is the purpose of this paper to describe the principle of operation, physical realization, and performance parameters for a reciprocal phase shifter of this latter type.

52 citations


Journal ArticleDOI
TL;DR: In this paper, the results of measurements of the optical phase variations on a short line-of-sight path through the turbulent atmosphere are presented, where a helium-neon laser (6328 A) propagates along a folded 50m path that is one arm of a Michelson interferometer, and its interference fringes are observed.
Abstract: We present the results of measurements of the optical phase variations on a short line-of-sight path through the turbulent atmosphere. A helium–neon laser (6328 A) propagates along a folded 50-m path that is one arm of a Michelson interferometer, and its interference fringes are observed. A measured time-lagged structure function, found by using Taylor’s hypothesis, compares well with the theoretical predictions of Tatarski for a broad range of effective separations, in contrast with earlier observations. Simultaneous measurements of Cn2 are made to facilitate comparison with theory. Values of the outer scale, deduced from temperature measurements at a different time at the same location, indicate excellent agreement with the spacing corresponding to the knee of the observed phase structure function, a result predicted by Tatarski’s theory and not previously observed.

52 citations


Patent
Ronald L Treadway1
12 Nov 1970
TL;DR: In this article, a digital frequency/phase detector employs a plurality of NAND gates interconnected to respond to changes in logic level of two input signals, the frequency and phase of which is to be compared.
Abstract: A digital frequency/phase detector employs a plurality of NAND gates interconnected to respond to changes in logic level of two input signals, the frequency/phase of which is to be compared. The detector is responsive to changes in the trailing edges of the input waveforms and produces outputs that are related to the repetition rate and relative phase of the inputs. The duty cycle of the input waveforms is unimportant since the circuit responds only to the trailing edge transitions in the input signal; and when the input signals are of the same frequency and are in phase, the output of the phase detector is a constant DC level.

52 citations


Journal ArticleDOI
D. H. Kelly1
TL;DR: In this article, the effect of truncating a low-frequency grating at a given phase is just opposite from the effect that truncating high-frequency gating at the same phase.
Abstract: When measured with small, sharp-edged targets, sinusoidal-grating thresholds are biased by two types of artifacts related to Weber’s and Ricco’s laws. These artifacts depend on both the nominal frequency of the grating and the phase at its edge: the effect of truncating a low-frequency grating at a given phase is just opposite from the effect of truncating a high-frequency grating at the same phase. These edge effects are predicted by spatial-frequency analyses of the targets, and confirmed by appropriate modulation-threshold measurements, with both rectilinear and circular gratings.

51 citations


Journal ArticleDOI
TL;DR: In this paper, the effects of beam modulation in the recording of a hologram were discussed and a unified description of fringe loci and fringe localization that is valid for arbitrary object motions and arbitrary beam modulation was presented.
Abstract: A comprehensive formulation of hologram interferometry is modified to include the effects of beam modulation in the recording of a hologram. The result is a unified description of fringe loci and fringe localization that is valid for arbitrary object motions and arbitrary beam modulation. This result is used to discuss the effect of beam modulation on fringe loci and localization conditions for a wide variety of cases: where one time function is sufficient to describe the motion of all points on an object surface; where a combination of independent time functions is required; where a combination of dependent time functions is required; where phase modulation proportional to the time function of the object’s motion is introduced; and where an object may execute sinusoidal vibrations that must be added as phase vectors (phasor vibrations). Localization conditions are shown to have potentially useful properties, for example, to permit separate observation of independent object motions when they occur simultaneously.

49 citations


Journal ArticleDOI
TL;DR: A means is demonstrated for determining the phase of vibrating objects by using either time average holography or real time holographic interferometry, which places an interferometric node (fringe-contrast maximum) at any object point.
Abstract: A means is demonstrated for determining the phase of vibrating objects by using either time average holography or real time holographic interferometry. The technique consists of vibrating a mirror in either the object or reference leg at the same frequency as the object and observing the effects on the fringe pattern when either the phase or amplitude of the mirror vibration is altered. Proper adjustment of these parameters places an interferometric node (fringe-contrast maximum) at any object point. The phase is then known from the phase of the vibrating mirror.

45 citations


Journal ArticleDOI
TL;DR: In this article, the authors present a closed form expression for best attainable resolution as a function of these same parameters, via the results of a phase error analysis by Brown and Riordan.
Abstract: Radar waves propagating through the turbulent troposphere acquire phase fluctuations which appear as aperture phase errors in synthetic-aperture radar systems. The phase error description follows directly from Tatarski's theory of wave propagation through turbulence; for most situations of interest, the phase error is Gaussian with stationary first increments. The structure function of the phase error is a function of the radar parameters and of the vertical profile Cnm(h) of the microwave refractive index structure constant of the turbulence. A closed form expression for best attainable resolution can then be given as a function of these same parameters, via the results of a phase error analysis by Brown and Riordan. Published data on Cnm(h) and its optical-region counterpart are reviewed. The data from a recent microwave propagation experiment by Thompson and Janes are then analyzed and are consistent with the order-of-magnitude estimates of Cnm: Some unsolved problems associated with applying the results to synthetic-aperture systems are discussed.

42 citations


Patent
07 Jan 1970
TL;DR: In this article, a digital waveform generator for providing digital waveforms particularly of the sinusoidal type was proposed, which comprises means for providing a sequence of digital numbers whose values vary in accordance with the phase versus time function of the waveform to be generated.
Abstract: A digital waveform generator for providing digital waveforms particularly of the sinusoidal type. The invention comprises means for providing a sequence of digital numbers whose values vary in accordance with the phase versus time function of the sinusoidal waveform to be generated. The function of time may be linear or nonlinear in accordance with the desired waveform. The sequence of numbers is utilized to address a digital memory wherein is stored the digital sinusoidal functional values corresponding to the digital phase values. As the sequence of phase numbers addresses the memory in accordance with the function of time, the corresponding sequence of sinusoidal numbers are provided by the memory, thus generating the desired digital waveform. Two embodiments are disclosed, one of which utilizes two registers for controlling the frequency and phase, respectively, of the output waveform. The other embodiment utilizes a difference equation computer to approximate the phase function by a recursively generated polynomial function of time.

Journal ArticleDOI
A. D. Wilson1
TL;DR: In this article, the fringes obtained by time-average holographic interferometry of a generalized time-dependent optical phase function are analyzed and the characteristic functions are sums of products of Bessel functions and exponential phase factors.
Abstract: This paper presents an analysis of the fringes obtained by time-average holographic interferometry of a generalized time-dependent optical phase function. The generalized optical phase function considered is the sum of a series of sinusoidal functions of time having arbitrary amplitudes, frequencies, and relative phases. Characteristic functions are determined for various optical phase functions of interest in time-average holography. In general, the characteristic functions are sums of products of Bessel functions (zero order and higher orders) and exponential phase factors. Rationally and irrationally related frequencies are included in this analysis. An example of vibrating string is considered, to illustrate the application of the results of this paper to objects vibrating at a multitude of frequencies.

Journal ArticleDOI
TL;DR: The work of Gary, Montgomery, and Swift (1968) is extended to show that a coherent electrostatic wave whose wave number is a slowly varying function of position will produce a nearly monoenergetic beam of particles.
Abstract: The work of Gary, Montgomery, and Swift (1968) is extended to show that a coherent electrostatic wave whose wave number is a slowly varying function of position will produce a nearly monoenergetic beam of particles The effect is demonstrated by numerical examples and by analysis of a phase plane diagram Numerical calculations for the motion of a particle in a wave whose phase is interrupted at regular intervals indicate that an incoherent wave can produce a burst of particles with a broad energy spectrum

01 Jan 1970
TL;DR: In this paper, the phase center is defined as the center of curvature of the intersection of a far-field phase surface with the horn axis, and the solution is therefore primarily applicable to sectoral horns.
Abstract: OR CERTAIN applicat.ions, for example, a very precise interferometer, it is necessary to know t.he phase center of a horn ant.enna and t.he dependence of the pha.se cent,er on the shape of the horn. The phase center is defined as the center of curvature of the intersection of a far-field phase surface wit.11 a plane containing the horn axis. Previous work in this a.rea has been done by Baur [I]. He bases his work on t.he Iiirchhoff a,pproximation [a], and he derives an elegant method for determining the phase center of the E plane. Because I

Journal ArticleDOI
TL;DR: It is shown that the power delivered onto a target and thus the return signal can be significantly increased by the principle of adaptive phase-distortion compensation, and that large arrays can be utilized in spite of the distorting effects of the atmosphere.
Abstract: A holographic technique to compensate for atmospherically induced phase distortion of a 106-micro laser beam is presented After a brief outline of the principle of adaptive phase-distortion compensation, the experimental setup to demonstrate feasibility is described Results obtained for a reflecting target at distances of 150 m and 4600 m are presented and discussed in detail It is shown that the power delivered onto a target and thus the return signal can be significantly increased by the principle of adaptive phase-distortion compensation By compensating for phase distortions in both the transmitted and received beams, the signal-to-noise ratio of the received signal can be improved by a factor of N(2), N being the number of apertures used, if the phase relation was completely random beforehand The results of these tests demonstrate that large arrays can be utilized in spite of the distorting effects which are normally produced by the atmosphere

Patent
16 Dec 1970
TL;DR: In this article, an electronic distance measuring device and method for determining the distance between two locations utilizing electromagnetic or light wave energy is presented. But the method is not suitable for the use of wireless communication.
Abstract: An electronic distance measuring device and method for determining the distance between two locations utilizing electromagnetic or light wave energy. The wave energy is transmitted from one location to the other and is reflected back toward the transmitting location. The reflected energy is detected and compared in phase with the transmitted energy, and the phase relationship between the transmitted and detected energy varied until a predetermined phase relationship exists. The predetermined phase relationship is selected so that the wave energy along the path between the transmitter, reflector and detector contains an integral number of wavelengths of wave energy. A time gate related to the propagation of the wave energy along this path is generated and distance is displayed digitally in response to the time gate and a highly stable reference frequency signal.

Journal ArticleDOI
TL;DR: For a linear uniform array of n elements, an expression for the directivity as a function of the spacing and the phase constants was derived in this article for the cases of isotropic elements, collinear short dipoles, and parallel short dipole.
Abstract: For a linear uniform array of n elements, an expression is derived for the directivity as a function of the spacing and the phase constants. The cases of isotropic elements, collinear short dipoles, and parallel short dipoles are included. The formula obtained is discussed in some detail and contour diagrams of the directivity as a function of the spacing and the phase constants in the above-mentioned cases are exhibited.

Journal ArticleDOI
T. Zachos1, J. Dyment
TL;DR: In this article, the propagation characteristics of junction-laser beams are described mathematically in a manner that is consistent with detailed beamwidth measurements in the Fresnel and Fraunhofer regions of the diffraction field.
Abstract: In Parts I and II of this work [1], [2], the resonant modes of GaAs lasers, as predicted from a proposed model, are found to be in very good agreement with experimental results that emphasize the frequency domain. This paper, Part III, supplements the previous work by investigating the spatial field distributions of junction-laser beams. For the first time, the propagation characteristics of the beams are described mathematically in a manner that is consistent with detailed beamwidth measurements in the Fresnel and Fraunhofer regions of the diffraction field. This correlation between theory and experiment is made possible by the use of a lens-slit combination that ensures the generation of reproducible beams with rectangular symmetry. The astigmatic beam solutions describing the propagation characteristics beyond the slit include beam-broadening effects produced both by the lens and by phase variations at the laser mirror, along the junction plane. These latter variations have not been previously reported. Their existence is deduced from our measurements beyond the slit, which show a greater beam expansion along the junction plane than can be theoretically accounted for by the lens system alone. Near threshold, at a distance of one beamwidth from the center of the mirror illumination, the phase lag is found to have a current-dependent magnitude of a few degrees.

Journal ArticleDOI
TL;DR: In this article, phase matching occurs when the crystal momentum of the acoustic phonon is adjusted to equal the familiar momentum mismatch observed in conventional second-harmonic generation in cubic GaAs.
Abstract: Radiation near 5.3 \ensuremath{\mu}m arising from the collinear phase-matched mixing of a 10.6-\ensuremath{\mu}m $Q$-switched laser beam and a pulsed ultrasonic shear wave has been observed in GaAs. Phase matching occurs when the crystal momentum of the acoustic phonon is adjusted to equal the familiar momentum mismatch observed in conventional second-harmonic generation in cubic GaAs. Three components of the susceptibility governing the four-wave interaction process were measured.

Patent
20 Feb 1970
TL;DR: In this article, a system for remotely controlling the phase of active antenna elements in a phased array that has the capability of correcting the inherent phase misalignment of each individual antenna element within the array is presented.
Abstract: A system for remotely controlling the phase of active antenna elements in a phased array that has the capability of correcting the inherent phase misalignment of each individual antenna element within the array. Beam azimuth and elevation signals are fed into an appropriate arithmetic unit by a radar signal processor. Roll, yaw and pitch signals are fed into the same arithmetic unit by an inertial navigation system. The arithmetic unit, using the information from the inertial navigation system and the radar signal processor, computes the phase shift required for each individual antenna element in order to point the radar beam in the desired direction. To compensate for inherent phase alignment in the system, an antennule position and alignment store feeds a predetermined alignment factor for each antennule into the arithmetic unit. The arithmetic unit then adds or subtracts the alignment factors from the respective computed phase angles. All the calculated phase angles are then sent to the control circuitry of the antenna array. A Csc2 beam phase shift store provides for generation of a Csc2 beam at the array.

Patent
B Bjorkman1
22 Dec 1970
TL;DR: In this paper, the phase of the intermediate frequency in the receiver corresponding to each transmitted frequency is measured, compared with a fixed phase reference oscillator and stored, and the stored phase information relating to each transmit frequency is used to control a phase shifter in the receivers during subsequent transmissions of the corresponding frequency, thereby completing a phase correction control loop.
Abstract: A communications device consisting of a transmitter that produces a phase modulated signal that is switched to a plurality of different frequencies according to a preselected program, and a receiver that mixes the received signal with a sequence of frequencies equal to the transmitted frequencies plus or minus a fixed intermediate frequency. In order to compensate for the different phase shifts produced by the interaction of the different transmitted frequencies with various fixed obstructions the phase of the intermediate frequency in the receiver corresponding to each transmitted frequency is measured, compared with a fixed phase reference oscillator and stored. The stored phase information relating to each transmitted frequency is used to control a phase shifter in the receiver during subsequent transmissions of the corresponding frequency, thereby completing a phase correction control loop. A similar comparison, storage, and control feedback loop is used in conjunction with an AVC amplifier in order to individually control the amplitude of each received frequency.

Journal ArticleDOI
29 Jan 1970
TL;DR: In this paper, an analytical procedure is developed to obtain the directional spectrum from records of an array of wave gages using two methods: the "locked phase method" and the "random phase method".
Abstract: The ocean surface may be considered to be composed of many waves traveling at different directions with different frequencies A graphical plot showing the allocation of wave energy to the different component frequencies and directions is the directional spectrum Directional spectrum has many applications in Coastal Engineering Herein an analytical procedure is developed to obtain the directional spectrum from records of an array of wave gages The two methods developed are the "locked phase method" and the "random phase method The locked phase method can be used to obtain the distribution of both phase as well as energy of the waves with respect to frequency and direction and is a deterministic approach The random phase analysis, on the other hand, is more suitable for wind waves in the ocean and yields just the distribution of energy alone as in most other procedures of spectrum analysis The procedures programmed for computers are checked using simulated data and laboratory data Wave records of the Pacific Ocean obtained off Point Mugu, California, on a 5-gage array were analyzed using the method developed and examples of the directional spectra obtained are presented.

Journal ArticleDOI
TL;DR: In this article, the coherence of the phase record and the space and time correlation functions of the ground diffraction pattern were derived from phase path changes with a resolution of less than 20 m. Incoherent phase records were most common but coherent records were often obtained.

Journal ArticleDOI
TL;DR: In this article, the phase aberration at position t across the aperture is assumed to have a power law structure function, E{[(?(t)-?(?)]2} = c|t-?|n.
Abstract: Resolution limits and corresponding optimum linear apertures are determined in the presence of phase errors. Let ?(t) be the phase aberration at position t across the aperture; it is assumed that the random process ? has a power law structure function, E{[(?(t)-?(?)]2}= c|t-?|n. Beam tilting caused by the phase error is "removed" (for each sample of ?), then resolution formulas are developed. An approximate analysis is obtained in closed form and yields an optimum resolution proportional to c1/n for O < n < 2. The exact analysis is given for Gaussian ?, and again the optimum resolution is proportional to c1/n. In applications n= 5/3 is of interest, and in the Gaussian case the best obtainable equivalent rectangle resolution is ? ?)/2? (0.975)c3/5 radians with a corresponding optimum linear aperture of 14c-3/5. When long exposures are considered, imaging without removing beam tilting is of interest, and resolution is degraded by a factor of about 2.5 for a linear aperture. Alternatively, in some applications optimum focus as well as beam tilt should be considered, in this case resolution is improved by a factor of about 1.4 (again for n= 5/3). Finally, joint (tilt corrected) optimization over aperture length and taper is treated; however, as one might expect, the use of taper offers negligible resolution improvement.

Journal ArticleDOI
TL;DR: A nomen­ clature is suggested which should allow a systematic classification of the various wavefront reconstruction processes and make clear the difference between the types of wavefront recordings.
Abstract: There are many different types of wavefront recordings in use having different mechanisms for reconstructing the waves. Some of these recordings are holograms in the original sense of the word in that all the information is recorded. Others deliberately throw away portions of the information. A system­ atic nomenclature is needed to clarify which technique is used and what information is involved. The greatest source of con­ fusion, at present, for those being introduced to holography is that of phase. For example, what are the differences between a phase hologram, a phase-only hologram, and a kinoform? This discussion was prompted by the confusion allowed by the present terminology. In addition to making clear the difference between the types of wavefront recordings, a nomen­ clature is suggested which should allow a systematic classification of the various wavefront reconstruction processes. A wavefront to be recorded and reconstructed can be described by the expression

Journal ArticleDOI
TL;DR: In this paper, the authors analyzed second-harmonic generation (SHG) with focused pump beams in the high-conversion regime where pump depletion was significant, and derived numerical solutions of the coupled equations for pump and harmonic fields for cylindrically symmetric case of collinear phase matching.
Abstract: We analyze second-harmonic generation (SHG) with focused pump beams in the high-conversion regime where pump depletion is significant. Numerical solutions of the coupled equations for pump and harmonic fields are obtained for the cylindrically symmetric case of collinear phase matching. The optimum values of phase mismatch and confocal parameters σ and b found by Boyd and Kleinman are very nearly independent of pump power to above 95 percent conversion. When phase mismatch is not optimized, the best confocal parameter depends upon the input power. The conversion efficiency versus input power is determined numerically for optimum \sigma, b , and approximate analytical formulas are given for this relation.

Journal ArticleDOI
TL;DR: In this paper, a technique for measuring the local phase speed in a turbulent flow is described, which is used to measure the phase speed of the Fourier components of the longitudinal velocity fluctuations in grid turbulence.
Abstract: A technique is described for measuring the local phase speed in a turbulent flow. The technique has been used to measure the phase speed of the Fourier components of the longitudinal velocity fluctuations in grid turbulence. These measurements are unique in that the probe spacing is only twice the Kolmogoroff length scale. The velocity fluctuations were measured with linearized constant-temperature hot-wire anemometers, the outputs of which were digitally sampled and recorded in real time. Digital Fourier analysis techniques were then used to calculate the cross-spectral density of the two velocity measurements. From this, the phase, phase speed, and coherence were calculated. The coherence has been used to estimate the variance of these measurements.

Proceedings ArticleDOI
11 May 1970
TL;DR: In this article, a low-loss X-band reciprocal latching ferrite phase shifter was proposed for low peak and average power levels in phased array radar systems, which can handle relatively high peak power levels and require no bias to maintain phase setting.
Abstract: Industry has given considerable attention to the development of low-cost phase shifters to be used as beam steering elements in phased array RADARs. Latching reciprocal ferrite phase shifters have received special attention because of the following considerations: (1) they can handle relatively high peak and average power levels, (2) they require no bias to maintain a phase setting, and (3) their reciprocal nature allows transmitting and receiving functions to occur without resetting the array. This paper presents a technique for obtaining a low loss X-band reciprocal latching ferrite phase shifter. The phase shifter is described by first considering the propagation constant of the circularly-polarized phase shift section and then discussing the nonreciprocal circular polarizers which are necessary to obtain reciprocal phase shift. The theoretical and experimental results of the phase shifter are then presented.

Journal ArticleDOI
TL;DR: In this paper, a method based on the measurement of the time-integrated irradiance obtained in an interference experiment is described, which is of particular utility in cases in which amplitude and phase or both are stochastic fluctuating variables.
Abstract: A method based on the measurement of the time-integrated irradiance obtained in an interference experiment is described. This method is of particular utility in cases in which amplitude and phase or both are stochastic fluctuating variables. When only phase fluctuations are present or when amplitude and phase are stochastic independent variables, simple relations can be derived from which the parameters of the distribution can be derived. In these cases, if a gaussian-distribution law for the phase fluctuations is assumed, the mean-square value of the phase fluctuations is directly measured. The method has been applied to the study of phase fluctuations of a laser beam, caused by atmospheric turbulence, with results that agree with the findings of other techniques. The anisotropy of turbulence can also be evaluated.

Journal ArticleDOI
TL;DR: In this article, the amplitude and phase of a pulsed wave scattered in the specular direction were measured at a frequency of 95.8 kHz for grazing angles from 6° to 80°.
Abstract: Three rough surfaces 32 in. by 32 in. in size were made of a pressure‐release material. The amplitude and phase of a pulsed wave scattered in the specular direction were measured at a frequency of 95.8 kHz for grazing angles from 6° to 80°. The theoretical approach of Gulin was used successfully in the Fresnel approximation to predict the observed amplitude and phase fluctuations for a Rayleigh parameter less than 0.6. Experimental measurements were extended to a Rayleigh parameter of 7.4. The observed fluctuations in the amplitude reached saturation at a Rayleigh parameter of 2.0 and fluctuated about this maximum as the parameter increased to 7.4. The fluctuations in the phase increased with the Rayleigh parameter, but there is some evidence that the phase fluctuation, like the amplitude fluctuations, saturates.