Suppression of higher diffraction orders in the extreme ultraviolet with a trapezoidal nano-mirror array

ABSTRACT

Higher diffraction orders from a grating introduce harmonic contamination in ‘monochromatic’ output beams processed by grating monochromators at synchrotron radiation facilities, resulting in imprecise calibration of optical elements and detectors. Suppressing these orders can be achieved with a quasi-random nano-hole array, but its fabrication can be a significant limiting factor for implementation. Here, an advanced type of grating is demonstrated that contains a periodic array of several hundred million trapezoidal nano-mirrors with a grazing-incidence reflection geometry that suppresses higher diffraction orders. Moreover, it reduces the difficulty of fabrication to the level of a traditional grating. It has great potential for harmonic suppression in synchrotron radiation, spectral diagnostics of plasmas, and astrophysics.     

Diffraction efficiency of volume gratings with finite size: Corrected analytical solution

Abstract

In order to correct errors in a previously published work on a treatment of two-dimensional coupled-wave theory, a complete derivation of the two-dimensional ‘overlap grating’ coupled wave equations is given. By using the Riemann method, a corrected solution to the equations in closed mathematical form is obtained. On the basis of this solution a brief investigation of the diffraction properties of finite-sized gratings, and in particular the dependence of diffraction efficiency on the geometric size of gratings, is given.     

Nonlinear Optical Rotatory Dispersion: Application of Group Theoretical Statistical Mechanics

Abstract

The third principle of group theoretical statistical mechanics is used to define the existence of many new nonlinear optical rotatory phenomena caused by the interaction of intense electromagnetic radiation with molecular ensembles. This treatment removes the necessity of having to distinguish between ‘natural’ and ‘magnetic’ optical activity; both processes emerge consistently from the analysis, which is based on an application of the Hellman-Feynman theorem and a double Taylor expansion of the induced electric and magnetic dipole moments.     

Quantum theory of light in nonlinear media with dispersion and absorption

Abstract

We present a canonical quantum theory of radiation in nonlinear media taking into account the effects of linear dispersion and absorption in a consistent way. We start from a microscopic model and apply recently developed concepts of the quantization of radiation in dispersive and absorbing linear media and extend the theory in order to include nonlinear optical processes. We derive propagation equations in space and time for the quantized radiation field, with special emphasis on the propagation of quantum-light pulses in Kerr media. In particular, the method enables us to derive systematically the noise sources that arise naturally in the nonlinear terms.     

Waveguide confinement of cerenkov second-harmonic generation through a graded-index grating coupler: Electromagnetic optimization

Abstract

The feasibility of making a frequency doubler for integrated optics is studied with the electromagnetic theory of gratings and graded-index waveguides as a tool. The device consists of a first waveguide filled with a sol-gel nonlinear material doped or grafted with a nonlinear chromophore whose thickness is chosen to generate a second-harmonic Cerenkov radiation in a dispersive glass substrate. The Cerenkov radiation is coupled into a second waveguide through a graded-index layer produced by ion exchange into glass, lying on top of an ion-etched grating coupler. The aim of the study is to optimize the optogeometrical parameters of the device in order to obtain a resonance line of the second waveguide modes that has an angular width large enough to match the experimental constraints, and which leads to a good enough coupling coefficient. The electromagnetic theory of grating couplers is developed into an S-matrix propagation algorithm form in order to be combined with the electromagnetic analysis of the thick graded-index waveguide with a view to analysing the device.     

High-sensitivity Diffraction-compensated Moiré Deflectometry

Abstract

The spatial resolution and angular sensitivity of conventional moiré deflectometry are known to obey a conjugate relationship reminiscent of the uncertainty principle so that one cannot be improved without deteriorating the other. This paper looks at the origin of this relationship and suggests ways by which this ‘uncertainty principle’ can be violated. It is shown that a ‘projection’ of the grating which is free from diffraction effects and which, at the same time, contains the deflection information amplified by the ‘optical lever’ distance D between the gratings can be formed if the diffracted beams of order + 1 and ? 1 are separated from other orders and are recombined in certain ways. As a result, the angular sensitivity, which depends on the ratio of the pitch d of the grating to the distance D between the two gratings, can be improved without losing the spatial resolution by multiple diffraction orders. One of the diffraction compensation schemes is experimentally illustrated. It is shown that the ‘uncertainty product’ is improved by more than two orders of magnitude.     

Commutation Rules and Eigenvalues of Spin and Orbital Angular Momentum of Radiation Fields

Abstract

We investigate the separation of the total angular momentum J of the electromagnetic field into a ‘spin’ part S and an ‘orbital’ part L. We show that both ‘spin’ and ‘orbital’ angular momentum are observables. However, the transversality of the radiation field affects the commutation relations for the associated quantum operators. This implies that neither S nor L are angular momentum operators. Moreover their eigenvalues are not discrete. We construct field modes such that each mode excitation (photon) is in a simultaneous eigenstate of S _z and L _z. We consider the interaction of such a photon with an atom and the resulting effect on the internal and external part of the atomic angular momentum.     

Beam Splitter Model of Two-beam Coupling in Photorefractive Materials

Abstract

The phenomenon of two-beam coupling in photorefractive crystals results from the interaction of two beams with the index grating written by those two beams. In this paper we show that the process can be understood by considering the photorefractive crystal to be a nonlinear beam splitter, with reflectivity and transmission coefficients dependent upon the relative intensities of the two beams. The relative phase relationships upon reflection, determined by the Fresnel equations, as well as the phase shift between the intensity and index grating, are the sources of the asymmetric coupling. A simple derivation of coupled mode equations describing two-beam coupling based upon the beam splitter model is proffered.     

On the scattering of light by a periodic structure in the presence of randomness II. On the detection of weak periodicities

Abstract

Light scattering by a rough grating is examined in the context of detecting a hidden periodic part. We make use of the fact that differentiated signals enhance the structures contained in the signal so that these derivatives help to obtain a priori ideas about the structure of the surface from which details about the surface profile can be worked out. The suggested numerical methods to extract the statistical properties of the surface and of the periodic grating involve simple algebraic operations and are similar to ‘matched filtering’ in signal detection problems. Numerical illustrations are provided to elucidate the method and to explain its usefulness. The suggested method, although simple, is seen to be an improvement vis-à-vis existing notions about detectability of periodic structures hidden behind a disorder.     

Stimulated emission of gamma-radiation: A proposed experiment

Abstract

The purpose of this paper is to describe a method for observing stimulated emission of gamma radiation. Stimulated emission of gamma radiation has never been observed. Stimulated emission of gamma radiation will be essential for developing gamma-ray lasers. The method uses the ‘π phase-shift induced transparency’ of gamma radiation, previously called the ‘gamma echo’. The gamma-echo technique is based on a modification of the time-differential Mössbauer spectroscopic method, i.e. the source is moved ‘instantaneously’ during the lifetime of the excited state. Then a resonant absorber becomes transparent to the resonant gamma radiation. There is a type of ‘self-stimulated’ emission in which the radiation ‘absorbed’, before the phase shift, is stimulated to emit after the phase shift. To observe stimulated emission of the resonant gamma radiation, one performs the same gamma-echo experiment except the resonant absorber is replaced by another source. This secondary source is chosen to be in resonance with the original source. Under these conditions there will be no stimulated emission, in the secondary source, before the π phase shift because the incident radiation is absorbed. However, after the phase shift, the secondary source becomes transparent to the incident radiation. If one uses a matched pair of Co⁵⁷ sources and the secondary source is 2mCi, there will be about 3 nuclei in the secondary source, in the first excited state of Fe⁵⁷, that are available to be stimulated during the 97.8ns lifetime of the primary source radiation.     

The Diffraction Efficiency of Double-grating Holographic Fan-out Elements

Abstract

An analytical expression is derived for the diffraction efficiency of a holographic fan-out element containing two superimposed gratings separated by a small angle. In addition to the usual zero and first orders an infinite set of the significant spurious waves is considered. In deriving the expression all these spurious waves are assumed to satisfy the Bragg condition exactly. The results are compared with a direct numerical solution and give very good agreement for angles up to 1°. Furthermore the analysis provides useful information for angles up to 5°. An analogy is drawn with a hybrid hologram consisting of a thick ‘carrier’ grating and a thin ‘modulation’ grating. Using this model the diffraction efficiencies can be calculated using the standard grating formulae. For fan-out applications the spurious waves around the first orders are considered the most important. From this analysis it can be seen that they can be made negligibly small but at the cost of a reduction in total diffraction efficiency.     

Second Harmonic Generation in Optical Fibres: Multiphoton Absorption and Translation-inversion Symmetry

Abstract

Multiphoton absorption provides a route to x ₍₂₎ grating formation in optical fibres. It explains the efficacy of ‘seeding’ the fibre with green light to induce fibre second harmonic generation. The process depends on absorption sites with spatially inverted configurations having different multiphoton absorption rates.     

Numerical simulation of a novel all-optical flip-flop based on a chirped nonlinear distributed feedback semiconductor laser structure using GPGPU computing

A novel all-optical flip-flop based on a chirped nonlinear distributed feedback laser structure is proposed. The flip-flop does not require a holding beam. The optical gain is provided by a current injection into an active layer. The nonlinear wave-guiding layer consists of a chirped phase shifted grating accompanied with a negative nonlinear refractive index coefficient that increases in magnitude along the wave-guide. In the ‘OFF’ state, the chirped grating does not provide the required optical feedback to start lasing. An optical pulse switches the device ‘ON’ by reducing the chirp due to the negative nonlinear refractive index coefficient. The reduced chirp grating provides enough feedback to sustain a laser mode. The device is switched ‘OFF’ by cross gain modulation. GPGPU computing allows for long simulation time of multiple SET–RESET operations. The ‘ON/OFF’ transitions delays are in nanoseconds time scale.     

A Model for the Optical Transistor and Optical Switching

Abstract

We propose a model for the optical transistor which can also be used for optical switching. We consider the ‘resonant optical tunnel effect’ and we show that the amplification of a modulation may be obtained by acting on the index of refraction of the intermediate nonlinear layer of a multilayer optical film. Parameters can be chosen in a technically possible range and we give a numerical estimation of these parameters when using CdS as amplifier. These results are compared with those which could be expected with a nonlinear Fabry-Perot.     

Optically induced non-radiative fast pulses in metals

Abstract

We describe new experiments in which the signals induced inside a solid conductor by external excitation using nanosecond long pulsed laser radiation are detected using a pyro-electric material. From the data obtained, it is shown that there is a previously unreported ‘early’ or ‘fast’ signal which propagates inside the solid at a speed greater than that reported for sound in the same media. This signal is observed in addition to the usually observed thermal and acoustic waves that travel at the speed of sound in the same media. We demonstrate that this signal cannot be adequately accounted for by existing theories of thermo-elastic wave propagation in solids.     

Continuous Wavelength Interferometry

A two-beam interferometer with a dispersive sample in its measuring beam is illuminated with quasi-monochromatic radiation of a grating monochromator. The wavelength u of this radiation is continuously changed through an extended wavelength range by turning the grating, and a dispersive continuous wavelength interferogram I (u) is recorded at a fixed point of observation. If the real index of refraction of the sample is known absolutely for only one observed wavelength u ₀, the dispersion of its complex index of refraction can be extracted out of the dispersive wavelength interferogram after subtracting a similar interferogram for the empty interferometer. This new and effective method of ‘scanning or continuous wavelength interferometry’ is fully described on the basis of some examples for the visible wavelength range.     

Calculation of Nonlinear Waves Guided by Optical Fibres Using the Resonance Technique

Abstract

A recently developed, simple, and easily programmable method based on a resonance technique for the analysis of nonlinear guided modes for planar geometry has been extended to include fibre configurations. We consider here nonlinear media of the general type (non-Kerr media) including the more complicated but realistic case of saturable media. In the present paper, as an example of the proposed methodology, the following configuration is treated: a circular optical fibre consisting of a linear core bounded by a nonlinear cladding medium. The numerical results obtained by the method, correspond to the ‘weak guidance’ approximation, including both field distribution and power calculations.     

Evanescent Wave Mixing on Nonlinear Surfaces

Abstract

The problem of using evanescent fields in nonlinear optics is discussed by employing results on quantization of evanescent waves. It is shown that the peculiar properties of the momentum of evanescent modes can be used to realize non-critical optical frequency mixing. In the first illustration, the case of surface second-harmonic generation is discussed. It is then shown that, in the case of difference-frequency generation, it is possible to generate a ‘completely evanescent mode’ which is ‘trapped’ by the surface, which becomes a two-dimensional waveguide.     

A modified approach for collimation testing using Lau interferometry

A modified approach for the collimation testing of an incoherent optical beam using the Lau effect coupled with moiré readout is proposed. In the experimental set-up a white-light incoherent source illuminates a coarse grating G_1. The collimating lens is mounted on a translation stage, so that the grating G₁ can be adjusted to be at the ‘in-focus’, ‘at-focus’ or ‘out-of-focus’ position. Grating G₂ and grating G₃ are mounted on a precision rotational stage and are so positioned that the lines of the gratings make a small but equal and opposite angle with the vertical. The inclination angle of the resulting moiré fringe demonstrates the quality of collimation of optical beam. The proposed technique can also be conveniently used for measurement of the focal length of a collimating lens, at a very low cost.     

ROBUSTNESS OPTIMIZATION FOR CONSTRAINED NONLINEAR PROGRAMMING PROBLEMS∗

In realistic situations engineering designs should take into consideration random aberrations from the stipulated design variables arising from manufacturing variability. Moreover, many environmental parameters are often stochastic in nature. Traditional nonlinear optimization attempts to find a deterministic optimum of a cost function and does not take into account the effect of these random variations on the objective. This paper attempts to devise a technique for finding optima of constrained nonlinear functions that are robust with respect to such variations. The expectation of the function over a domain of aberrations in the parameters is taken as a measure of ‘robustness’ of the function value at a point. It is pointed out that robustness optimization is ideally an attempt to trade off between ‘optimality’ and ‘robustness’. A newly-developed multi-criteria optimization technique known as Normal-Boundary Intersection is used to find evenly-spaced points on the Pareto curve for the ‘optimality’ and ‘robustness’ criteria. This Pareto curve enables the user to make the trade-off decision explicitly, free of arbitrary ‘weighting’ parameters.

This paper also formulates a derivative-based approximation for evaluating the expected value of the objective function on the nonlinear manifold defined by the state equations for the system. Existing procedures for evaluating the expectation usually involve numerical integration techniques requiring many solutions of the state equations for one evaluation of the expectation. The procedure presented here bypasses the need for multiple solutions of the state equations and hence provides a cheaper and more easily optimizable approximation to the expectation. Finally, this paper discusses how nonlinear inequality constraints should be treated in the presence of random parameters in the design. Computational results are presented for finding a robust optimum of a nonlinear structural optimization problem.