Complex Amplitude Of Field Research Articles

Superresolution imaging from limited-aperture optical diffracted field data

The use of an optical waveguide to attain a numerical aperture of unity in computational coherent optical imaging applications is described. It is shown that for the case of a one-dimensional (slitlike) object radiating into an optical waveguide consisting of two plane-parallel mirrors the complex field amplitude across any cross section of the waveguide contains sufficient information to reconstruct the object's transmittance function with a numerical aperture of unity. We include the derivation of an inversion algorithm for performing the object reconstruction as well as computer simulations of the procedure.

Analysis of the interferometric effect of the background light in apertureless scanning near-field optical microscopy

We investigate in detail the interferometric nature of the signal delivered by an apertureless scanning near-field optical microscope (SNOM). This nature is first brought to the fore by near-field images of an integrated waveguide. The detection process of an evanescent wave generated by total internal reflection is then studied by both lateral near-field scans and signal detection as a function of the tip-to-sample distance. This study permits interpretation of fringe patterns appearing in apertureless SNOM images and provides important information about the nature of the signal. In particular, both experimental data and simple calculations show that, because of interference with background light coming from the sample, the detected signal can describe the complex field amplitude, or the field intensity, or a subtle mix of both, depending on the tip environment and the tip position.

Terahertz beam propagation measured through three-dimensional amplitude profile determination

To determine the spatio-temporal field distribution of freely propagating terahertz bandwidth pulses, we measure the time-resolved electric field in two spatial dimensions with high resolution. The measured, phase-coherent electric-field distributions are compared with an analytic model in which the radiation from a dipole antenna near a dielectric interface is coupled to free space through a spherical lens. The field external to the lens is limited by reflection at the lens–air dielectric interface, which is minimized at Brewster’s angle, leading to an annular field pattern. Field measurements compare favorably with theory. Propagation of terahertz beams is determined both by assuming a TEM0,0 Gaussian profile as well as expanding the beam into a superposition of Laguerre–Gauss modes. The Laguerre–Gauss model more accurately describes the beam profile for free-space propagation and after propagating through a simple optical system. The accuracy of both models for predicting far-field beam patterns depend upon accurately measuring complex field amplitudes of terahertz beams.

The ray approach for analyzing the modal structure of the sound field in a range-dependent waveguide

Simple ray-approximation formulas are obtained for the mode amplitudes in a waveguide with range-dependent parameters. The idea of the proposed approach is based on the mode expansion of the complex field amplitude determined using the geometrical-optics approximation. A specific example of calculating mode amplitudes is analyzed for a deep-water sound channel with a sound speed profile nonadiabatically varying with distance. The results of the calculation are compared with the numerical solution obtained for the same problem by the parabolic equation method.

Theory of a multimode semiconductor laser with optical feedback

We discuss the derivation of multimode rate equations for the description of a semiconductor laser with external cavity. We adopt a formulation where the complex field amplitudes are coupled to the nonlinear gains. For N lasing modes, this leads to $2N$ equations that display in-phased and antiphased time-dependent solutions. A simplified reference model is obtained by assuming that the key parameters are frequency independent. A general linear stability analysis leads to the prediction of two types of Hopf bifurcations. A nondegenerate Hopf bifurcation occurs with the relaxation oscillation frequency as the characteristic bifurcation frequency. A $(N\ensuremath{-}1)$-degenerate Hopf bifurcation occurs with a lower characteristic frequency. To assess the nature and stability of the solutions emerging from the Hopf bifurcations, we perform a nonlinear stability analysis on a reduced model obtained in the limit of large linewidth enhancement factor. In this asymptotic limit, the steady state is always destabilized in favor of a stable periodic inphased or antiphased state. A numerical analysis yields a bifurcation diagram of the multimode equations, which confirms the analytic results and reveals further complex regimes (quasiperiodic and chaotic, in-phased and antiphased) as the amplitude of the field fed back into the laser is increased.

Talbot array illuminators providing spatial intensity and phase modulation

For the first time to our knowledge, Talbot array illuminators are investigated in terms of providing specific periodic spatial phase patterns at distances showing maximum contrast in intensity. As the main outcome of this investigation, we present a new class of array illuminators, i.e., phase grating profiles, exhibiting an alternating 0–π phase distribution at fractional Talbot distances. The new phase profiles were obtained by means of numerical evaluations of the near field's complex amplitude distribution with the Fresnel–Kirchhoff diffraction formula behind spatially finite grating structures. We further discuss the relationship between spatial phase modulation at distances of maximum contrast and the increase in the fundamental frequency of the intensity distribution compared with that of the input phase structures.

Chaotic behavior in transverse-mode laser dynamics.

We analyze chaotic behavior found in numerical simulations of the transverse pattern dynamics of a laser demonstrating that in some cases chaos originates in phase dynamics and is of low dimension. Investigations of both a Ginzburg-Landau equation for the complex field amplitude of the laser output and a Kuramoto-Sivashinsky-type equation for only the phase of that complex field equation find the same behavior. Both equations can be expanded in terms of spatial modes and in the chaotic regime the behavior of the modal amplitudes seems relatively independent. However, the fluctuations of the modal amplitudes are sufficiently correlated so that the spatiotemporal dynamics is a form of low dimensional chaos rather than a more complex turbulent behavior or even one that might merit the term spatiotemporal chaos.

Coherent properties of focused wide partially coherent beams in inhomogeneous media

The quasioptics approximation is used to analyze the propagation of a partially coherent radiation beam in a statistically inhomogeneous medium with a regular refractive index profile. This involves solving the parabolic equation for the complex field amplitude which in particular makes it possible to generalize the van Cittert–Zernike theorem (propagation of radiation in inhomogeneous media). The coherence radius is calculated and its ratio to the transverse size of the beam is analyzed.

Filament formation in semiconductor laser gain regions

We have linearized the equations for propagation of the beam of light in a semiconductor optical amplifier about an operating point and have derived the rate of growth of small sinusoidal perturbations of the phase and modulus of the complex field amplitude. The perturbations grow if the spatial frequency is below a critical value that depends on the intensity of the field at the operating point. For spatial frequencies above the critical value, the perturbations die out. The critical spatial frequency decreases as the intensity increases above a certain value. In other words, the tendency to filament becomes weaker as the intensity increases above a certain value. Computer-generated solutions of the propagation and gain equations are included to illustrate the growth of filaments as the plane-wave intensity changes in an amplifier.

The high-gain FEL equation: an inexpensive numerical algorithm

In this paper we discuss a simple and efficient algoritm which allows reliable solutions of the FEL small-signal high-gain equation, even with the pocket computer. We use the same algorithm to calculate the derivatives of the complex field amplitude, with respect to the detuning parameter. The interest, for these last quantities, is motivated by the crucial role they play in the analysis of the evolution of the FEL transverse and longitudinal modes.

Stochastic adiabatic approximation and laser fluctuation

The stochastic adiabatic approximation developed recently by the authors is generalized to treat systematically the laser fluctuations. We predict the effect of two time scales of the intensity correlation function in some non-dye lasers. The laser field amplitude equation we obtained possesses several interesting properties due to the fact that the laser field is driven by a new type of noise. It is shown that the physical origin of the difference of the statistical properties between the colored loss-noise model and the colored gain-noise model is a feature of the intensity dependent correlation time of the multiplicative noises in the complex field amplitude equation. The relations between our results and the previous stochastic theories of the single-mode laser are discussed.

Problems for the elimination of atomic variables and the single-mode dye laser equations

The stochastic generalization of the adiabatic approximation has been developed earlier by Wunderlin and Haken [6] and recently by Schoner and Haken [7] and the authors [8]. Using these theories, a complex field amplitude equation for the single-mode dye laser is derived based on a set of semiclassical single-mode laser equations. An effective Fokker-Planck equation in which the laser intensity is decoupled from the phase variable is obtained. The relations between our theory and the phenomenological single-mode dye laser equation are discussed.

Propagation of a VLF electromagnetic wave packet in a magnetoplasma

In this paper the propagation of a VLF electromagnetic wave packet in a magnetoplasma is studied. The wave packet is constructed by superposing a group of spectrum‐weighted characteristic vectors. Accurate to the second order, a set of analytic expressions of the complex amplitude factor and field vectors for a Gaussian packet has been obtained. Based on these expressions, effects of dispersion and anisotropy on the vector property of wave field and the amplitude envelope are simulated. Numerical results show that the polarization of wave fields is varying when the packet moves and somewhat deviates from exact circular polarization in a plane transverse to the carrier wave vector, ko. The Gaussian envelope is found to rotate around the Vg×ko direction and to spread three dimensionally as it propagates. The rotation of the ellipsoidal pulse tends toward a stable direction after a propagation of about 250 periods.

A new single-mode laser equation

A new complex field amplitude equation for the single-mode laser with several colored multiplicative noises is derived. The relation between our equation and the single-mode dye laser equations with gain noise is discussed. To study the statistical properties of the single-mode laser using the equation we obtained, an effective Fokker-Planck equation (EFPE) in which the laser intensity is decoupled from the phase variable is derived.

Efficient second harmonic conversion of broad-band high-peak-power Nd:glass laser radiation using large-aperture KDP crystals in quadrature

The authors have investigated the second-harmonic conversion efficiency of broadband Nd:glass laser light ( Delta nu /c<30 cm/sup -1/ FWHM). Using two KDP crystals in a quadrature arrangement, they obtain energy conversion efficiencies of approximately=55% with an initial bandwidth for the fundamental of Delta nu /c approximately=17 cm/sup -1/ FWHM. For these conditions, a modest increase ( approximately=70%) was observed in the harmonic bandwidth (FWHM) relative to the fundamental. The usual theory of three-wave mixing in dispersive birefringent nonlinear crystals is extended to describe the broadband harmonic conversion process; the generalized theory includes the statistical properties of the light and phase mismatch effects on the spectral components in the complex field amplitudes. Good agreement is found between the code calculations and the measurements.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exact quantum statistics of a nonlinear dissipative oscillator evolving from an arbitrary state.

The statistical properties of the third-order nonlinear dissipative oscillator, which evolves from any state, are derived on the basis of the exact solution to the master equation. Some important features of the nonlinear oscillator model, such as the recurrences of the initial state, are related to the properties of quasidistributions connected with the phase of the complex field amplitude.

Integrable and chaotic polarization dynamics in nonlinear optical beams

The problem of two counterpropagating optical laser beams in a nonlinear medium is investigated as a Hamiltonian system. The phase space for travelling-wave solutions is the manifold C2 × C2, coordinated by two complex two-compnent electric field amplitudes, one for each beam. Invariance of the Hamiltonian function under various actions of the rotation group allows for reduction of the phase space to the two-sphere S2, on which the reduced Hamiltonian system, being two-dimensional, is completly integrable. We determine all the fixed points of the system and describe the bifurcations of the phase portrait which occur as parameters are varied. Among the various orbits in the reduced phase space, those connecting the hyperbolic fixed points are special and correspond to soliton-like and kink-like travelling-wave solutions. We also investigate how chaos, in particular horseshoe chaos and Arnold diffusion, arises when the system is subjected to certain types of perturbations.

A microwave fel code using waveguide modes

A free electron laser code, GFEL, is being developed for application to the LLNL Microwave Tokamak Experiments, MTX. This single frequency code solves for the slowly varying complex field amplitude using the usual wiggler-averaged equations of existing codes, in particular FRED [1], except that it describes the fields by a 2D expansion in the rectangular waveguide modes, using coupling coefficients similar to those developed by Wurtele [2], which include effects of spatial variations in the fields seen by the wiggler motion of the particles. Our coefficients differ from those of Wurtele in two respects. First, we have found a missing √2 γaw factor in his Cz; when corrected this increases the effect of the Ez field component and this in turn reduces the amplitude of the TM mode. Second, we have consistently retained all terms of significant order in the wiggle amplitude. Both corrections are necessary for accurate computation. GFEL has the capability of following the TEmn and TMmn modes simultaneously. GFEL produces results nearly identical to those from FRED if the coupling coefficients are adjusted to equal those implied by the algorithm in FRED. Normally, the two codes produce results that are similar but different in detail due to the different treatment of modes higher than TE01.

Wigner-function Description of Quantum Noise in Interferometers

Abstract A quantum-statistical description of noise in interferometers is given in terms of the Wigner distribution function. The interferometer may contain an amplifier in one of its arms. The input field can be in a variety of states such as a Fock state, a coherent state or a squeezed coherent state. The Wigner function of the output field at the detector is shown to have a general Gaussian form with non-zero complex field amplitude and with other parameters related to the characteristics of the input field, amplifier and beam splitters, etc. An explicit form of the photon-number distribution is given. Higher-order correlations of the output field can be obtained from the Gaussian property of the Wigner function.

Nonlinear dynamics and spectral behavior for an external cavity laser

The influence of nonlinear dynamics on the linewidth, spectral behavior, and stability properties for a semiconductor laser in an external cavity is examined by numerical simulation of the noise-driven rate equations. Experimental results on the feedback-induced linewidth reduction are presented. The saturation of linewidth reduction and the sudden increase in linewidth which is observed at high feedback levels are shown to be contained in the theoretical model. The state with strongly increased linewidth has the signature of a chaotic state. We present examples of the calculated time evolution of the complex field amplitude and the associated FM noise spectra and field power spectra. The examples illustrate the transition to the chaotic state and focus on the spectral characteristics of the state.