Fundamental Fourier Component Research Articles

Aharonov–Bohm oscillations, quantum decoherence and amplitude modulation in mesoscopic InGaAs/InAlAs rings

Low-temperature Aharonov–Bohm oscillations in the magnetoresistance of mesoscopic interferometric rings patterned on an InGaAs/InAlAs heterostructure are investigated for their dependence on excitation current and temperature. The rings have an average radius of 650 nm, and a lithographic arm width of 300 nm, yielding pronounced interference oscillations over a wide range of magnetic fields. Apart from a current and temperature dependence, the oscillation amplitude also shows a quasi-periodic modulation with applied magnetic field. The phase coherence length is extracted by analysis of the fundamental and higher Fourier components of the oscillations, and by direct analysis of the amplitude and its dependence on parameters. It is concluded that the Thouless energy forms the measure of excitation energies for quantum decoherence. The amplitude modulation finds an explanation in the effect of the magnetic flux threading the finite width of the interferometer arms.

Transmission of colour and acuity signals by parvocellular cells in marmoset monkeys

The red-green axis of colour vision evolved recently in primate evolutionary history. Signals serving red-green colour vision travel together with signals serving spatial vision, in the parvocellular (PC) division of the subcortical visual pathway. However, the question of whether receptive fields of PC pathway cells are specialized to transmit red-green colour signals remains unresolved. We addressed this question in single-cell recordings from the lateral geniculate nucleus of anaesthetized marmosets. Marmosets show a high proportion of dichromatic (red-green colour-blind) individuals, allowing spatial and colour tuning properties of PC cells to be directly compared in dichromatic and trichromatic visual systems. We measured spatial frequency tuning for sine gratings that provided selective stimulation of individual photoreceptor types. We found that in trichromatic marmosets, the foveal visual field representation is dominated by red-green colour-selective PC cells. Colour selectivity of PC cells is reduced at greater eccentricities, but cone inputs to centre and surround are biased to create more selectivity than predicted by a purely 'random wiring' model. Thus, one-to-one connections in the fovea are sufficient, but not necessary, to create colour-selective responses. The distribution of spatial tuning properties for achromatic stimuli shows almost complete overlap between PC cells recorded in dichromatic and trichromatic marmosets. These data indicate that transmission of red-green colour signals has been enabled by centre-surround receptive fields of PC cells, and has not altered the capacity of PC cells to serve high-acuity vision at high stimulus contrast.

Motion of the surface of the human tympanic membrane measured with stroboscopic holography

Sound-induced motion of the surface of the human tympanic membrane (TM) was studied by stroboscopic holographic interferometery, which measures the amplitude and phase of the displacement at each of about 40,000 points on the surface of the TM. Measurements were made with tonal stimuli of 0.5, 1, 4 and 8kHz. The magnitude and phase of the sinusoidal displacement of the TM at each driven frequency were derived from the fundamental Fourier component of the raw displacement data computed from stroboscopic holograms of the TM recorded at eight stimulus phases. The correlation between the Fourier estimates and measured motion data was generally above 0.9 over the entire TM surface. We used three data presentations: (i) plots of the phasic displacements along a single chord across the surface of the TM, (ii) phasic surface maps of the displacement of the entire TM surface, and (iii) plots of the Fourier derived amplitude and phase-angle of the surface displacement along four diameter lines that define and bisect each of the four quadrants of the TM. These displays led to some common conclusions: at 0.5 and 1kHz, the entire TM moved roughly in-phase with some small phase delay apparent between local areas of maximal displacement in the posterior half of the TM. At 4 and 8kHz, the motion of the TM became more complicated with multiple local displacement maxima arranged in rings around the manubrium. The displacements at most of these maxima were roughly in-phase, while some moved out-of-phase. Superposed on this in- and out-of-phase behavior were significant cyclic variations in-phase with location of less than 0.2 cycles or occasionally rapid half-cycle step-like changes in-phase. The high frequency displacement amplitude and phase maps discovered in this study can not be explained by any single wave motion, but are consistent with a combination of low and higher order modal motions plus some small traveling-wave-like components. The observations of the dynamics of TM surface motion from this study will help us better understand the sound-receiving function of the TM and how it couples sound to the ossicular chain and inner ear.

The partial Talbot effect and its use in measuring the coherence of synchrotron X-rays

The Talbot effect is the self-imaging, at distances D multiple of D(R), of the intensity downstream of a periodic object. Earlier work with hard synchrotron radiation X-rays showed the variation with D of the fundamental Fourier component of intensity to be a good measurement of beam coherence. Any higher-order Fourier coefficients I (D, m > 1) would be periodic with a reduced period D(Rm) = D(R)/m for an ideally coherent incident beam (partial Talbot effect). The degree of coherence gamma(x) is sampled through the ratio of I (D, m) at D = 0 and multiples of D(Rm). This requires the Fourier coefficient for D = 0, which is not accessible for a phase object (no contrast at D = 0). However, the ratio of the slopes of I (D, m) at D = 0 and D = pD(Rm) also provides this information. Furthermore, a characterization of gamma(x) is possible, provided an assumption is made on its shape, using only the ratio of the Fourier coefficient I (D, m) of two images a distance pD(Rm) apart. Experiments with one- and two-dimensional phase gratings and a mixed (amplitude and phase) two-dimensional grating confirm that the partial Talbot effect approach is viable. It requires a reduced range of distances, and yields important results directly, obviating the need for computer fits. In particular, 8% of the beam intensity was found to have very low coherence in the vertical direction, probably due to monochromator imperfection.

The photorefractive effect at large modulation depth in semiconductors with multiple defect levels

The photorefractive effect in semiconducting materials with multiple defects is studied in the case of modulation depth m=1. The basic equations are Poisson’s equation and the continuity equations for electrons, holes and occupied defect levels. They include all recombination and optical generation mechanisms between the defect levels and valence and conduction bands. Their explicit numerical solution yields microscopic quantities such as space- and time-dependent electrical field profiles, carrier concentrations, as well as generation and recombination rates. The fundamental Fourier component of the electric field yields the two-wave-mixing gain. Application is made for InP with two levels in the forbidden gap, for which steady-state and transient resulting quantities are shown. The resulting features at large modulation depth are of non-sinusoidal shape. Due to the complexity of the system, the final results strongly depend on all parameters intervening in the models used, as is illustrated for several typical cases.

Effects of bicuculline on direction-sensitive relay cells in the dorsal lateral geniculate nucleus (LGNd) of cats

The direction sensitivity of relay cells in the cat’s dorsal lateral geniculate (LGNd) was measured using sinusoidal grating stimuli before and during local bicuculline administration. One hundred and twenty-eight LGNd relay cells were recorded in laminae A and A1, of which 44 relay cells (34%) were found to be sensitive to direction of stimulus movement. The direction-sensitive LGNd relay cells could be differentiated into two subgroups based on different measures of their response amplitude. Type I cells exhibited their direction sensitivity when the fundamental Fourier component (FFC) of the poststimulus time histograms (PSTHs) was used as response measure, but did not show significant direction sensitivity when mean firing rate was used. Type II cells exhibited their direction sensitivity, no matter whether the FFC or mean firing rate was used as the measure. Of 35 cells analyzed, 27 cells remained direction sensitive during bicuculline administration. At the population level, the direction bias of type I cells did not change systematically, while the direction bias of type II cells decreased significantly during bicuculline administration. These results suggest that the direction bias of these two types of relay cells are mediated by different neural mechanisms. The direction bias of type I cells may involve multiple inputs from spatio-temporally separate subunits within retinal ganglion cells receptive fields. The direction bias of type II cells may involve GABAergic neuronal circuits within the LGNd.

Visual motion of missing-fundamental patterns: motion energy versus feature correspondence

Missing-fundamental gratings, generated by subtracting the fundamental Fourier components from square-wave gratings, appear to move backward when presented in quarter-cycle jumps, even though their edges and features all move forward. We used variants of these stimuli to test current models of motion perception. We found that missing-fundamental plaids, constructed from orthogonal missing-fundamental gratings, also appear to move backward. Forward motion was restored to missing-fundamental gratings and plaids by adding back small fractions of the original fundamental. In-phase and antiphase addition of the fundamental had similar effects on the perceived motion, despite having markedly different effects on the features, appearances and zero-crossings of the stimuli. The critical amplitude of fundamental needed to restore forward motion to plaids was the same as that needed to restore forward motion to their isolated component gratings, indicating that the plaids’ emergent features, such as edge intersections and ‘blobs’, made little or no contribution to the perceived direction of motion in these stimuli. In two derivative experiments, missing-fundamental chromatic gratings and plaids, at approximate isoluminance, and missing-fundamental luminance barberpoles, also generated backward perceived motions, and these were also reversed by in-phase or antiphase addition of small amounts of fundamental.

The missing-fundamental illusion at isoluminance.

The missing-fundamental illusion describes how a square wave with its fundamental Fourier component removed appears as a square wave. This illusion is normally explained with reference to the bandpass nature of the luminance-contrast-sensitivity function, together with a 'default-to-square-wave' rule. Since the chromatic-contrast-sensitivity function is low-pass, we should not expect a missing-fundamental illusion at isoluminance. Using a simultaneous-detection-and-identification paradigm to eliminate contrast as a cue to discrimination, we nevertheless found that chromatic missing fundamentals and square waves could not be separately identified at detection threshold: just under twice the contrast required to detect the stimuli was needed to identify them. To test whether this was due to insufficiently narrow chromatic-channel bandwidths, we measured detection and identification thresholds for chromatic F and 3F sine-wave gratings. In this case identification was possible almost at detection threshold, suggesting that channel bandwidth limitations were not the critical factor. It is suggested that the weak missing-fundamental illusion observed at isoluminance probably reflects the operation of mechanisms similar to those that are responsible for the chromatic Craik-Cornsweet illusion.

The salt finger amplitude in unbounded T-S gradient layers

Finite amplitude numerical calculations are made for a completely unbounded salt e nger domain whose overall vertical ‘ ‘ property’ ’ gradients ( Tz andSz) are uniform and remain unaltered in time. For diffusivity ratio t 5 k S/k T 5 O(1), Prandtl number n /k T ae 1, and density ratio R 5 Tz/Sz . 1 this regime corresponds to a ‘ ‘ double gradient’ ’ sugar ( S) 2 salt (T) experiment. Two-dimensional pseudo-spectral calculations are made in the vicinity of the minimum critical condition for salt e nger instability, viz., small e ; (Rt ) 2 1 2 1 . 0; the allowed spectrumincludes the fastest growing wave of linear theory. When the vertical wavelength of the fundamental Fourier component is systematically increased the solution changes from a single steady vertical mode to a multi-modal statistically steady chaotic state. Each of the long vertical modes can be amplie ed by the (unchanging overall) gradient Sz, and can be stabilized by the induced vertical T, S gradients on the same scale as the modes; nonlinear triad interactions in the T 2 S equations can also lead to amplitude equilibration even though e , k T/n , and the Reynolds number are extremely small. When subharmonics of the horizonal wavelength of maximum growth are introduced into the numerical calculations the new wave amplie es (via Sz) and produces a quantitative change in the time average e uxes. Experimentally testable values of heat e ux and rms horizontal T-e uctuations are computed in the range2.8 . R . 1.6 for t 5 1 A3.Asymptotic similarity laws e ® 0 are also presented.

Analysis, modeling, and simulation of series-parallel resonant converter circuits

This paper presents a SPICE macromodel for a generic series-parallel resonant converter circuit. The model is derived from the averaged time-invariant state-space equations obtained from a Fourier transform. The conditions are derived under which all but the fundamental harmonic may be discarded, and the model developed based solely on the fundamental Fourier component. The single macromodel developed has a wide range of validity, and allows DC, AC, and transient analyses to be carried out in a fast, easy, and familiar manner. It also permits the converter to be incorporated alongside its control circuitry into an entire system. The simulation results from the model have been compared to results from a full simulation, and the agreement is found to be excellent, with the macromodel simulation running between 37 and 4700 times faster than the full simulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parallel resonant converter as a circuit simulation primitive

The paper presents a SPICE macromodel for a generic parallel resonant converter circuit. The model is derived from the averaged time-invariant state-space equations obtained from Fourier transform. The conditions are derived under which all but the fundamental harmonic may be discarded, and the model developed based solely on the fundamental Fourier component. The single macromodel developed has a wide range of validity, allowing DC, AC and transient analyses to be carried out using the entire converter as a circuit primitive. Thus, such a converter may be incorporated (alongside its control circuitry), as a primitive, into an entire system. Example analyses show a parallel resonant circuit, embedded in a feedback loop, becoming unstable when the load undergoes a step change. A similar set of analyses, carried out at component level, yield identical results, but require runtimes greater by a factor of between 192 and 6600. Finally, the &apos;stability region&apos; of the circuit, characterised by the load resistance, is calculated. This kind of information is extremely useful to the designer, and not usually easily obtained.

Variability of responses to sinusoidal modulation.

Many studies of visual neurons make use of stimuli that are sinusoidally modulated in time, and take as the response the fundamental Fourier component of the firing. This is a study of the variability of the fundamental sinusoidal components. A theoretical analysis shows that the variance of sinusoidal components should be nearly independent of their amplitudes; this is expected despite the observation that variance of firing rate increases with increasing firing rate. However, this result applies only to the variance of the complex amplitude, defined as the complex Fourier amplitude in response to each stimulus cycle. This variance is called the complex variance. The variance of the scalar amplitude, which is simply the amplitude in response to each stimulus cycle disregarding phase (scalar variance) is expected to shrink by a factor of up to 2 1/3 as the response magnitude approaches zero. If the relationship between variance of rate and rate is linear, complex variance should be independent of amplitude. If the relationship between variance of rate and rate is characterized by a compressive nonlinearity (as has been observed), the complex variance should very slightly decrease with increased amplitude, despite the main trend of increased variance of rate with increased rate. Data from cat ganglion cells stimulated with sinusoidally modulated lights of various contrasts agree with the theory, although some individual cases show trends that may be indicative of nonlinearity in the relationship between variance of rate and rate.

Effects of spatial filtering and lack of effects of visual imagery on pattern-contingent color aftereffects.

Checkerboards contain fundamental two-dimensional Fourier components oriented 45 degrees from the edges of individual checks. Previous studies have shown that contingent color aftereffects following adaptation to chromatic checkerboard stimuli were associated with the fundamental components rather than the edges. In the present experiments, we measured contingent color aftereffects, using the method of constant stimuli, after subjects adapted to unfiltered checkerboards and checkerboards with the fundamental Fourier components removed. The adaptation stimuli were magenta (or green) squares and green (or magenta) diamonds; the test stimuli were vertical or oblique sine-wave gratings with different saturations. After adaptation to unfiltered checkerboards, aftereffects contingent on the fundamental components were obtained. In contrast, after adaptation to filtered stimuli, aftereffects of smaller magnitude were found to be aligned with the edges. The data support the previous findings of spatial-frequency-contingent color aftereffects with checkerboard adaptation stimuli and indicate that the aftereffects can be associated with edges if the fundamental components of adaptation stimuli are removed by spatial filtering. We reexamined the possibility of color aftereffects induced by imagery of checkerboards. Contrary to the previous reports, no significant aftereffects were obtained.

Frequency analysis of time-interval-modulated switched networks

A Fourier analysis of the time-interval-modulated switched networks is undertaken using time-varying system theory. The result is a linearized describing function approach which determines the small-signal control-to-output (fundamental Fourier component) frequency response. The proposed algorithm is general and exact, and the results are developed in closed form. The intended application area is that of switching DC-to-DC converters, where use of the algorithm is demonstrated in deriving an exact analytical expression for the control-to-output transfer function of pulse-width-modulated converters. Experimental results are presented, verifying the modeling technique. >

Hopf bifurcation in chemical kinetics

We study Hopf bifurcations in chemical reaction systems for their potential use in quantitative experimental analysis of the kinetics of oscillatory reactions. Three models of the Belousov–Zhabotinsky reaction are investigated as examples. For these we have determined and characterized the sub- and supercritical Hopf bifurcations in their dependence on parameters of the models. For supercritical bifurcations we calculate a number of parameters that can be used for quantitative comparison of models and experiment. In particular, we calculate expansion coefficients of the flow rate, the frequency of oscillation, and a Floquet exponent; the small parameter of the expansions is the square of the amplitude of the fundamental Fourier component of the oscillations. We also calculate quenching amplitudes from an adjoint eigenvector of the Jacobi matrix of the kinetics. They determine the conditions under which the small amplitude oscillations can be quenched by addition of the species participating in the reaction. The model properties are compared with experiment. They show qualitative, but not quantitative agreement.

Light adaptation and frequency transfer properties of cat horizontal cells

The frequency transfer properties of horizontal cells in the cat retina were studied as a function of the mean light intensity level and stimulus contrast. To this end, horizontal cell responses to sinusoidally modulated light stimuli were recorded intracellularly in the optically intact, in vivo eye. The light stimulus consisted of a 3.9 deg dia. spot superimposed on a steady background (8.8 deg dia.). A discrete Fourier analysts was performed in order to describe the amplitude and phase characteristics of the linear response component and in order to specify the nonlinear distortion of the response. The amplitude of the fundamental Fourier component was found to increase linearly with the contrast of the sinusoidal light intensity modulation. Increasing the mean light level while keeping the contrast constant caused a frequency dependent increase in response amplitude. The increase was most pronounced at high temporal frequencies and resulted in a conspicuous increase of the flicker fusion frequency. Steady background illumination caused a reduction of the response amplitudes at the lower temporal frequencies. Responses in the high frequency range, however, were not affected. The phase shifts of the fundamental Fourier components were found to diminish at increasing mean illumination levels. The harmonic distortion of horizontal cell responses to sinewave flicker was studied as a function of stimulus frequency and stimulus contrast. By comparing the data obtained using sinusoidal light intensity modulation with the intensity profiles described in a preceding paper it was investigated to what extent the harmonic distortion can be explained by the nonlinearity expressed in the response vs intensity profiles.

A numerical study of two-dimensional crystal growth forms in the presence of anisotropic growth kinetics

A numerical model of the free growth of a two-dimensional single crystal of pure material from an undercooled melt is presented. Quasi-steady-state heat flow and anisotropic interfacial properties couple in the model, governing the morphology of the evolving crystal-melt interface. We study the evolution of an initially slightly perturbed circular crystal as it grows from the linear into the non-linear regime both in the absence and in the presence of anisotropic interfacial growth kinetics. In the absence of kinetic anisotropy, a first harmonic of opposite phase is generated by the fundamental component and splitting of the tip of a perturbation occurs. For an initial shape consisting of a fundamental Fourier component and its first harmonic, there is strong interaction of the initial first harmonic with the first harmonic that would have been generated by the fundamental component acting along. In the absence of kinetic anisotropy, the first harmonic generated by a given Fourier component during growth adopts a phase, for given perturbation amplitudes, that appears to maximize the perimeter of the crystal. The overall rate of growth of crystal area is shown to be only a function of the heat extracted from the system and is virtually independent of the interface shape; this is a consequence of the quasi-steady state approximation to heat flow. Finally, we show that anisotropic growth kinetics can suppress tip splitting for certain initial conditions.

Nonlinearities in binocular visual evoked potentials in children.

Evidence is controversial in respect to the optimal conditions in which visual evoked potentials provide an objective measure of binocular visual function, related and unrelated to stereopsis, and there is little emphasis on the type of stimulus that produces facilitation in binocular recording. We investigated the effects of stimulus type (flicker or pattern), contrast, and temporal modulation on facilitation, which was defined as a binocular response greater than sum of monocular responses. Monocular and binocular responses to sinusoidally modulated flicker and grating patterns were recorded in children and Fourier analyzed. The relationship of the fundamental Fourier component for flicker and the second harmonic component for pattern were each examined as function of temporal modulation at two levels of contrast for monocular and binocular visual evoked potentials. Binocular facilitation was found across all conditions for flicker. Data suggest that processing of pattern and flicker has different sites of origin within the visual system. Facilitation in binocular visual evoked potentials also indicates that they are not a result of simple summation of monocular responses, since there appears a nonlinear component to such interaction.

Dynamical conversion of optical modes in garnet films induced by ferrimagnetic resonance

It is demonstrated that ferrimagnetic resonance can be applied to induce a dynamical conversion of optical modes in a magnetic garnet film. The fundamental and first-harmonic Fourier components of the converted mode are measured as a function of frequency and bias field; a dynamical conversion efficiency of 2% is obtained. The amplitude of the first harmonic depends very critically on the orientation of the magnetization with respect to the film normal. The results are confirmed by calculations on the basis of Faraday rotation and Cotton–Mouton effect.

Electroretinograms evoked by sinusoidal excitation of human cones.

The amplitude and phase of the fundamental Fourier component of the human electroretinogram (e.r.g.) were recorded with a synchronous detection method under conditions in which each of the three species of cones can be assumed most sensitive in turn. Weber-Fechner behaviour is well established at, or more distal in the retina than, the source of these voltages. Results over the frequency range 7-50 Hz exclude a diffusion model of human flicker perception. The e.r.g. phase vs. frequency plot found with a 'red' test differs from that obtained with a 'green'. The shapes of the e.r.g. field sensitivity action spectra agree with those of the subject's IIj(mu) (j = 3, 4 and 5) mechanisms of Stiles and with in situ measurements of the absorbance spectra of human cone pigments. Threshold phase with each test was independent of background wave-length but, consistent with the results in 3, the phase of the response to the 'red' test (25 Hz) differed significantly from that to the 'green'. If these differences resulted from the absorption of test photons of different colours at different points along the outer segment (independent of cone spectral sensitivity), they would be as clear on dichromats as on trichromats. Results on a protanope are inconsistent with this prediction. We infer that differences in phase are due to the different kinetics of different cone mechanisms and that the e.r.g.s studied here are evoked by exciting only the most sensitive cone mechanism, even though dark-adaptation studies prove that at e.r.g. threshold the test is well above psychophysical threshold for all three cone species. If the inference in 5 is correct, studies of sensitivity across the retina suggest that the spatial distribution of long-, middle- and short-wave-sensitive cones in the human retina differ remarkably.