Chirp Function Research Articles

A Parabolic Waveform Generator Based on the Chirp Characteristics of a Directly Modulated Laser

Due to carrier dynamics, the modulated light field from a directly modulated laser (DML) has an intensity envelope with a certain frequency chirp. When the chirp is linearly mapped into intensity by a frequency discriminator such as an optical filter with a linear edge, the optical field presents a new signal determined by the multiplication operation between the envelope function and the chirp function. Under a triangular drive signal, this process can contribute dark, bright and frequency-doubled bright parabolic waveforms by properly adjusting the filter window. This method is verified by both a theoretical analysis and experimental demonstrations. It not only provides a low-cost and simple scheme to generate parabola signals, but also a new method for arbitrary waveform generation.

Supercontinuum generation by saturated second-order nonlinear interactions

We propose a new approach to supercontinuum generation and carrier-envelope-offset detection based on saturated second-order nonlinear interactions in dispersion-engineered nanowaveguides. The technique developed here broadens the interacting harmonics by forming stable bifurcations of the pulse envelopes due to an interplay between phase-mismatch and pump depletion. We first present an intuitive heuristic model for spectral broadening by second-harmonic generation of femtosecond pulses and show that this model agrees well with experiments. Then, having established strong agreement between theory and experiment, we develop scaling laws that determine the energy required to generate an octave of bandwidth as a function of input pulse duration, device length, and input pulse chirp. These scaling laws suggest that future realization based on this approach could enable supercontinuum generation with orders of magnitude less energy than current state-of-the-art devices.

Exploring the performance of implicit neural representations for brain image registration

Pairwise image registration is a necessary prerequisite for brain image comparison and data integration in neuroscience and radiology. In this work, we explore the efficacy of implicit neural representations (INRs) in improving the performance of brain image registration in magnetic resonance imaging. In this setting, INRs serve as a continuous and coordinate based approximation of the deformation field obtained through a multi-layer perceptron. Previous research has demonstrated that sinusoidal representation networks (SIRENs) surpass ReLU models in performance. In this study, we first broaden the range of activation functions to further investigate the registration performance of implicit networks equipped with activation functions that exhibit diverse oscillatory properties. Specifically, in addition to the SIRENs and ReLU, we evaluate activation functions based on snake, sine+, chirp and Morlet wavelet functions. Second, we conduct experiments to relate the hyper-parameters of the models to registration performance. Third, we propose and assess various techniques, including cycle consistency loss, ensembles and cascades of implicit networks, as well as a combined image fusion and registration objective, to enhance the performance of implicit registration networks beyond the standard approach. The investigated implicit methods are compared to the VoxelMorph convolutional neural network and to the symmetric image normalization (SyN) registration algorithm from the Advanced Normalization Tools (ANTs). Our findings not only highlight the remarkable capabilities of implicit networks in addressing pairwise image registration challenges, but also showcase their potential as a powerful and versatile off-the-shelf tool in the fields of neuroscience and radiology.

Riesz transform associated with the fractional Fourier transform and applications in image edge detection

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of its connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform and is given by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transform, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.

Free-electron interactions with van der Waals heterostructures: a source of focused X-ray radiation

The science and technology of X-ray optics have come far, enabling the focusing of X-rays for applications in high-resolution X-ray spectroscopy, imaging, and irradiation. In spite of this, many forms of tailoring waves that had substantial impact on applications in the optical regime have remained out of reach in the X-ray regime. This disparity fundamentally arises from the tendency of refractive indices of all materials to approach unity at high frequencies, making X-ray-optical components such as lenses and mirrors much harder to create and often less efficient. Here, we propose a new concept for X-ray focusing based on inducing a curved wavefront into the X-ray generation process, resulting in the intrinsic focusing of X-ray waves. This concept can be seen as effectively integrating the optics to be part of the emission mechanism, thus bypassing the efficiency limits imposed by X-ray optical components, enabling the creation of nanobeams with nanoscale focal spot sizes and micrometer-scale focal lengths. Specifically, we implement this concept by designing aperiodic vdW heterostructures that shape X-rays when driven by free electrons. The parameters of the focused hotspot, such as lateral size and focal depth, are tunable as a function of an interlayer spacing chirp and electron energy. Looking forward, ongoing advances in the creation of many-layer vdW heterostructures open unprecedented horizons of focusing and arbitrary shaping of X-ray nanobeams.

Inequalities for the Windowed Linear Canonical Transform of Complex Functions

In this paper, we generalize the N-dimensional Heisenberg’s inequalities for the windowed linear canonical transform (WLCT) of a complex function. Firstly, the definition for N-dimensional WLCT of a complex function is given. In addition, the N-dimensional Heisenberg’s inequality for the linear canonical transform (LCT) is derived. It shows that the lower bound is related to the covariance and can be achieved by a complex chirp function with a Gaussian function. Finally, the N-dimensional Heisenberg’s inequality for the WLCT is exploited. In special cases, its corollary can be obtained.

Wakefield Generation and Enhancement of Electron Energy to the Multi-GeV in a Magnetized Plasma

The nonlinear interaction of an intense chirped laser pulse with transversely magnetized plasma is studied theoretically. The work described in this investigation is dealing with the characteristics of the wakefield generation by different chirping functions. Numerical results reveal that the amplitude and wavelength of the generated wakefield are highly influenced by the chirping constant and types of chirping function. Moreover, for the linear chirping function the positive chirping constant and for the nonlinear chirping functions the negative chirping constants excite larger wake amplitudes. In addition, it was found that by choosing a proper chirping constant for each chirping functions, one could obtain highly GeV electron energy gain. The maximum energy is obtained for the electron injected in the wakefield induced by the Exp and Linear chirped pulse. The maximum electron energy is about 2.5 GeV for the wake excited by the negative Exp chirp function. Further results revealed that the energy of the electron is significantly enhanced using an external transverse magnetic field.

Large-area periodically-poled lithium niobate wafer stacks optimized for high-energy narrowband terahertz generation.

Periodically-poled lithium niobate (PPLN) sources consisting of custom-built stacks of large-area wafers provide a unique opportunity to systematically study the multi-cycle terahertz (THz) generation mechanism as they are assembled layer-by-layer. Here we investigate and optimize the THz emission from PPLN wafer stacks as a function of wafer number, pump fluence, pulse duration and chirp, wafer separation, and pump focusing. Using 135 µm-thick, 2"-diameter wafers we generate high-energy, narrowband THz pulses with central frequencies up to 0.39 THz, directly suitable for THz-driven particle acceleration applications. We explore the multi-cycle pulse build-up with increasing wafer numbers using electro-optic sampling measurements, achieving THz conversion efficiencies up to 0.17%, while demonstrating unique control over the pulse length and bandwidth these sources offer. Guided by simulations, observed frequency-dependence on both stack-mounting and pump focusing conditions have been attributed to inter-wafer etalon and Gouy phase-shifts respectively, revealing subtle features that are critical to the understanding and performance of PPLN wafer-stack sources for optimal narrowband THz generation.

Approximation Theorems Associated with Multidimensional Fractional Fourier Transform and Applications in Laplace and Heat Equations

In this paper, we establish two approximation theorems for the multidimensional fractional Fourier transform via appropriate convolutions. As applications, we study the boundary and initial problems of the Laplace and heat equations with chirp functions. Furthermore, we obtain the general Heisenberg inequality with respect to the multidimensional fractional Fourier transform.

Integration of chirping and apodization of Topas materials for improving the performance of fiber Bragg grating sensors

The discovery of the fiber Bragg grating (FBG) is an early milestone in developing optical fiber technology, such as optical communication to monitoring material health structures as sensors. For optical communication, the FBG components are capable of filtering functions. As a sensor, it has a high sensitivity immune to electromagnetic wave interference, is small in size, and is resistant to extreme environmental conditions. The sensitivity of the FBG sensor is obtained from the shift in the peak wavelength of each of the temperature and strain quantities. However, the performance of the FBG sensor can be improved by engineering the distribution of the refractive index on the grid with the apodization and chirp functions. Apodization is a technique to improve the performance of the FBG to eliminate noise, narrow the full width half maximum, lower the side lobes of the main lobe, and improve the spectrum ripple factor. Apart from apodization, the chirp function also affects the sensor sensitivity and the refractive index distribution on the grid. Numerical experiments were carried out in designing the FBG component as a sensor using Gaussian apodization and Topas (cyclic olefin copolymer) for several chirp functions. The results show that the Gaussian apodization Topas for all chirp functions as a strain sensor has the same sensitivity, namely 0.84 pm/μstrain while for temperature sensors with the highest sensitivity is obtained at cubic root chirp of 13.82857 pm/°C followed by square root chirp of 13.74286 pm/°C, quadratic chirp 13.71429 pm/°C, and linear chirp 13.4 pm/°C. The Bragg wavelength shift was greater for 1 °C than for the 1 μstrain.

Optimal Design of Thinned Array Using a Hybrid Genetic Algorithm

In this paper, a hybrid genetic algorithm (GA) is proposed for thinning a two-dimensional planar array by combining the conventional GA with moving least squares (MLS). This enhances the convergence rate and the global search performance. The MLS method is used to estimate local interpolation functions from non-uniform sample data (the population and the value of the objective function in the GA), and to find new and better populations from the interpolated functions. By incorporating these improved populations into the next generation, the MLS-GA achieves improved search performance of the global optimum and a faster convergence rate compared to conventional GA alone. Moreover, a nonlinear chirp function is used for an efficient thinning design. To verify the proposed MLS-GA, it is applied to a test function and the results are compared to that of the GA. The algorithm is then applied to thin an array with a rectangular grid and circular boundary. The design objectives are to minimize the peak side-lobe level and gain loss while satisfying a given thinning coefficient and to compare the results with the GA.

Define and measure the dimensional accuracy of two-photon laser lithography based on its instrument transfer function

Two-photon polymerization (TPP) is a powerful technique for direct three-dimensional (3D) micro- and nanomanufacturing owing to its unique ability of writing almost arbitrary structures of various materials. In this paper, a novel method is proposed to quantitatively define and measure the dimensional accuracy of TPP tools based on the concept of instrument transfer function (ITF). A circular linear-chirp pattern is designed for characterizing the ITF. Such a pattern is arranged in a linear chirp function with respect to its radial distance from the pattern centre, thus, well representing a signal with a quasi-flat amplitude over a given spectral bandwidth. In addition, the pattern is rotational symmetric, therefore, it is well suited for characterizing the ITF in different angular directions to detect angular-dependent asymmetries. The feasibility of the proposed method is demonstrated on a commercial TPP tool. The manufactured pattern is calibrated by a metrological large range atomic force microscope (AFM) using a super sharp AFM tip, thus the dimensional accuracy and angular-dependent anisotropy of the TPP tool have been well characterized quantitatively. The proposed method is easy to use and reveals the dimensional accuracy of TPP tools under real manufacturing conditions, which concerns not only the optical focus spot size of the exposing laser but also influencing factors such as material shrinkage, light–matter interaction and process parameters.

A Nonlinear Hanning-Windowed Chirplet Model for Ultrasonic Guided Waves Signal Parameter Representation

Parametric signal representation techniques with Chirplet models have attracted much attention in ultrasonic guided waves-based research of material property identification and structural integrity evaluation. The known Chirplet models are established using either the Gaussian windows or the linear chirp function. This is inconsistent with practical situations where the excitation is a Hanning-windowed sinusoidal signal, and the received signals have a nonlinear phase and asymmetric envelope due to the dispersive nature of the waves. A nonlinear Hanning-windowed Chirplet (NHWC) model was proposed to eliminate the above inconsistencies in which a nonlinear phase modulation term was designed to modulate the classical Hanning-window and the sine function. The phase modulation term was established with the hyperbolic tangent function. The mathematical properties of the nonlinear phase modulation term and the NHWC model were analyzed mathematically, including the time variability, parity, concavity, and convexity of the phase. These were used to guide the parameter setting in parametric signal representation techniques. The NHWC model can characterize various characteristics of guided waves signals, including the symmetric or asymmetric Hanning envelopes and the phase nonlinearity. Finally, an adaptive genetic algorithm was adopted to verify the effectiveness of the NHWC model in the parameter representation of experimentally measured ultrasonic signals.

Basic Functions of Fiber Bragg Grating Effects on the Optical Fiber Systems Performance Efficiency

Abstract The study presents the basic apodization and chirp functions based fiber Bragg grating (FBG) for upgrading optical fiber performance efficiency. Variations of apodization and chirp functions are studied with grating length variations. Optical power after FBG, signal to noise ratio, maximum Q-factor, and output power are measured in the presence of the chirp functions. Gaussian apodization function has outlined good performance than other proposed apodization functions. As well as linear chirp function is better performance than other chirp functions. The optical system performance tested with/without chirp effects with high bit rates.

Performance Signature of Optical Fiber Communications Dispersion Compensation Techniques for the Control of Dispersion Management

Abstract This paper has deeply simulated the different configurations of pre, post, and symmetric dispersion compensation-based dispersion compensated fiber (DCF), Gaussian optic filter (GOF), and fiber Bragg grating. Output power, signal quality factor, and signal gain are the performance parameters in this study for the complete comparison for the proposed techniques for dispersion compensation. It is observed that FBG has presented the best performance parameters optimization in compared to other compensation techniques for different configurations of compensation. The FBG dispersion compensation technique is the key solution for dispersion compensation in compared with other compensation techniques that are comparatively as in previous section. It is observed that the FBG is the best dispersion compensation technique and change its location, and length, apodization, and chirp functions, to obtain the best performance of FBG and give the best quality factor, output power, and gain, FBG can be employed in different categories such as post compensation, pre compensation, and symmetric compensation with different FBG lengths, apodization, and chirp functions.

Fast τ-p transforms by chirp modulation

We have developed simple, fast, and accurate algorithms for the linear Radon ([Formula: see text]-[Formula: see text]) transform and its inverse. The algorithms have an [Formula: see text] computational complexity in contrast to the [Formula: see text] cost of a direct implementation in 2D and an [Formula: see text] computational complexity compared to the [Formula: see text] cost of a direct implementation in 3D. The methods use Bluestein’s algorithm to evaluate discrete nonstandard Fourier sums, and they need, apart from the fast Fourier transform (FFT), only multiplication of chirp functions and their Fourier transforms. The computational cost and accuracy are thus reduced to that inherited by the FFT. Fully working algorithms can be implemented in a couple of lines of code. Moreover, we find that efficient graphics processing unit (GPU) implementations could achieve processing speeds of approximately [Formula: see text], implying that the algorithms are I/O bound rather than compute bound.

Chirp-control of resonant high-order harmonic generation in indium ablation plumes driven by intense few-cycle laser pulses.

We have studied high-order harmonic generation (HHG) in an indium ablation plume driven by intense few-cycle laser pulses centered at 775 nm as a function of the frequency chirp of the laser pulse. We found experimentally that resonant emission lines between 19.7 eV and 22.3 eV (close to the 13th and 15th harmonic of the laser) exhibit a strong, asymmetric chirp dependence, with pronounced intensity modulations. The chirp dependence is reproduced by our numerical time-dependent Schrödinger equation simulations of a resonant HHG by the model indium ion. As demonstrated with our separate simulations of HHG within the strong field approximation, the resonance can be understood in terms of the chirp-dependent HHG photon energy coinciding with the energy of an autoionizing state to ground state transition with high oscillator strength. This supports the validity of the general theory of resonant four-step HHG in the few-cycle limit.

Modified Chirp Waveforms-Based OCC-UWB System With Multiple Interferences Suppression

To improve the performance of the ultra-wideband (UWB) system against multiple interferences existed in wireless personal area network (WPAN), such as narrowband interference (NBI), multiple-access interference (MAI), and multipath interference (MI), we present an orthogonal complementary code (OCC)-based UWB system with modified chirp waveforms. Considering the time–frequency characteristics of the chirp functions and the principle of NBI suppression, we first propose the modified linear and nonlinear chirp function-based flexible UWB pulses to suppress the multiple NBIs. Furthermore, with the implementation of OCC spread modulation, the modified chirp waveform-based UWB system is presented to effectively alleviate the influence of the MAI and MI. Simulation results and analysis validate the efficiency of the modified chirp pulse-based OCC-UWB system in the suppression of NBIs, MAI, and MI.

Nonresonant Raman Effects on Femtosecond Pump-Probe with Chirped White Light: Challenges and Opportunities.

Impulsive Raman excitation in neat organic liquids far from resonance is followed using chirped broad-band supercontinuum probe pulses. Spectral modulations due to impulsively induced coherent vibrations vary in intensity 10-fold as a function of the probe's linear chirp. Simulations clarify why the vibrational signature is maximized for a group delay dispersion (GDD) in reduced units of νvib-2 = 0.5 while a probe GDD of twice that quenches the same spectral modulations. Accordingly, recent claims that chirped white-light probe pulses provide equivalent information on material response to their compressed analogues must be taken with caution. In particular, interactions that induce spectral shifts in the probe depend crucially on the arrival chronology of the continuum colors. On one hand, this presents limitations to application of chirped continuum radiation as-is in pump-probe experiments. It also presents the opportunity for using this dependence to control the relative amplitude of nonresonant interactions in pump-probe signals such as that of solvent vibrations.

Ambiguity function and accuracy of the hyperbolic chirp: comparison with the linear chirp

In this study, the authors derive the ambiguity function (AF) of a narrowband and a wideband hyperbolic chirp. They calculate the second derivatives of the squared amplitude of the narrowband complex AF and use them to calculate the Fisher information matrix (FIM) of the estimators of the target range and velocity. The FIM is then used to calculate the Cramer-Rao lower bounds (CRLBs) of the variance of the estimators and to carry out an analysis of estimation performance and a comparison with the case of a linear chirp with a rectangular and a Gaussian amplitude modulation. The analysis and the calculations of the CRLB are also extended to a train of hyperbolic chirps. Results corroborate that at narrowband the hyperbolic chirp is less Doppler tolerant than the linear chirp and show that the hyperbolic chirp provides a comparable measurement accuracy to the linear chirp. Results at wideband corroborate the superior Doppler tolerance of the hyperbolic chirp with respect to that of the linear chirp.