Focal Shift Effect Research Articles

The Fresnel Approximation and Diffraction of Focused Waves

In this paper, diffraction of scalar waves by a screen with a circular aperture is explored, considering the incidence of either a collimated beam or a focused wave, a historical review of the development of the theory is presented, and the introduction of the Fresnel approximation is described. For diffraction by a focused wave, the general case is considered for both high numerical aperture and for finite values of the Fresnel number. One aim is to develop a theory based on the use of dimensionless optical coordinates that can help to determined the general behaviour and trends of different system parameters. An important phenomenon, the focal shift effect, is discussed as well. Explicit expressions are provided for focal shift and the peak intensity for different numerical apertures and Fresnel numbers. This is one application where the Rayleigh–Sommerfeld diffraction integrals provide inaccurate results.

Do Laguerre–Gaussian beams recover their spatial properties after all obstacles?

It has been known for some time that Bessel–Gaussian beams and their associated families self-heal after encountering obstacles. Recently, it has been shown theoretically and experimentally that radial mode Laguerre–Gaussian (LG) beams would likewise self-heal. In this work, we show that the self-healing occurs after opaque disks but not after circular apertures. We put forward arguments to explain this and perform key experiments, assessing the impact of obstructions on the usual intensity reconstruction, as well as the focal shift effect and the beam propagation factor, $$M^2$$ . In addition, these results are supported by a physical interpretation based on the Abbe experiment of spatial frequency filtering, highlighting that the self-healing process requires a high-spatial frequency component to the field, i.e., LG beams self-heal when modulated by a low spatial frequency filter (opaque disk) but not when modulated by a high-spatial frequency filter (circular aperture). We believe that this work will contribute significantly to the field of laser beam propagation through obstacles.

Estimation of time of tracking for designed absorber used in domestic solar desalination system using experimental and Finite Element analysis (ANSYS)

Domestic solar desalination system consists of manual tracking parabolic solar concentrator, absorber and condenser unit. Salt water kept in an absorber gets heated by solar rays focused on to it by using parabolic solar concentrator. Water gets converted into steam and then it is condensed in a condenser to get purified water without using electricity. Efficiency of the system can be improved by maximum utilization of solar energy, which depends largely on to the geometry of absorber and the time for tracking. Paper explains selection of size of absorber using experimental and Finite Element analysis. Size and the time at which bright spot remains on an absorber are measured and tracing of the incident rays is estimated using experimental analysis. Validation of analysis is made using ray tracing software and ANSYS.FEA analysis is made to find the effect of focal shift on surface temperature of absorber. Maximum time, for which solar energy retains on to an absorber without tracking, is also measured. Effect of variation in the sizes of absorber and focal shift is calculated and validated using experimental analysis. Quality of purified potable water is tested at different heights and positions in the water testing lab

Positive focal shift of gallium nitride high contrast grating focusing reflectors

We design a type of metasurfaces capable of serving as a visible-light focusing reflector based on gallium nitride (GaN) high contrast gratings (HCGs). The wavefront of the reflected light is precisely manipulated by spatial variation of the grating periods along the subwavelength ridge array to achieve light focusing. Different from conventional negative focal shift effect, a positive focal shift is observed in such focusing reflectors. Detailed investigations of the influence of device size on the focusing performance, especially the focal length, are preformed via a finite element method . The results show that all performance parameters are greatly affected by the reflector size. A more concentrated focal point, or a better focusing capability, can be achieved by larger size. With increasing reflector size, the achieved focal length decreases and gradually approaches to the design, thus the corresponding positive focal shift decreases. Our results are helpful for understanding the visible-light control of the planar HCG-based focusing reflectors.

Intensity Distributions of a Supercontinuum Laser in an Apertured Dispersion Lens**Supported by Science and Technology Planning Project of Hunan Province under Grant No 2013SK3144, Scientific Research Fund of Hunan Provincial Education Department under Grant No 11A011, the Construct Program of Key Discipline in Hunan Province, and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of

We present intensity distributions of a supercontinuum laser passing through an apertured dispersion lens and find focal shift effects generated. The results show that, apart from the conventional negative focal shift, a positive focal shift also appears in the supercontinuum laser. The maximum intensity of the supercontinuum laser shifts toward the lens when the truncated parameter is small. However, it exceeds the focus of the central wavelength and a positive focal shift appears in the supercontinuum laser with large truncated parameter. Both of the maximum intensities shift away from the focus when the bandwidth of the supercontinuum laser increases, while the shift direction is opposite. No focal shift appears when the bandwidth and the truncated parameter satisfy some conditions.

Reduction of Focal Shift Effects in Industrial Laser Beam Welding by Means of Innovative Protection Glass Concept

Abstract High-power laser beam welding in industrial environment often suffers from process induced contamination of laser focusing optics. Especially exposed to this contamination is the plane protection glass which is positioned directly above the process to protect the expensive lenses from contaminations such as spatter and metal vapor. Locally increased absorption due to con-tamination leads to a temperature rise in the protection glass and a corresponding change of its optical characteristics. This results in a reduced beam quality and a shift of the focus position. Both effects lead to a reduced intensity of radiation on the workpiece causing a lower welding penetration depth. In this article we present laser beam measurements using laser processing optics with protection glasses of different materials and different grades of contamination. Welds in mild steel illustrate the extraordinary advantage of sapphire protection glasses, allowing a constant welding depth even when they are strongly contaminated. Welding results, beam caustic measurements and an estimation of economic efficiency will be shown.

Positive and negative focal shifts of an apertured supercontinuum laser with rectangular spectrum

In this paper, diffraction of the supercontinuum beam from aperture and lens are described for the first time. We study the positive and negative focal shift effects of the supercontinuum laser on focusing system in detail. It is found that, apart from the conventional negative focal shift, the positive focal shift is also generated in the supercontinuum laser. The maximum intensity of the focused supercontinuum laser is shifted toward the lens with small truncated parameter. When the truncated parameter is large, however, the maximum intensity exceeded the focus of the central wavelength and the positive focal shift appears. The results also show that the maximum intensities are shifted away from the focus when the bandwidth of the supercontinuum laser increases in the two cases, but the shift direction is opposite.

Systematic study of the focal shift effect in planar plasmonic slit lenses

In this paper, we systematically studied the focal shift effect in planar plasmonic slit lenses. Through theoretical derivations and numerical simulations, we found that there is a focal length shift between the traditional design model and the finite-difference time-domain simulations. The shift is not only dependent on the Fresnel number (FN) of the lens, like traditional dielectric lenses, determined by the lens width and the designed focal length, but also on the surface plasmon polariton (SPPs) interaction on the lens surfaces, dependent on the slit numbers. We also found that the FN-induced focal shift is predominant when FN < 1. However, the SPP interaction-induced focal shift plays a major role when FN > 1. An approximated theoretical model is presented to estimate the focal shift of plasmonic slit lens with FN < 1.

Theory and implementation of focal shift of plasmonic lenses

We utilize three diffraction theories to investigate the focal-shift phenomena of plasmonic lenses. By taking the Fresnel number into account, a better accuracy in predicting the relative focal shift can be achieved for low-Fresnel-number systems. On the basis of this focal-shift effect, we present a method to realize an active plasmonic lens. Two-dimensional finite-difference time-domain simulation is performed to verify the method proposed.

Focal shift of flat-topped beams passing through a lens system with or without aperture

We theoretically investigated on the focal shift of flat-topped beams (FTBs) passing through a lens system with or without aperture. We find that the position of on-axis peak intensity of focused FTBs strongly depends on the Fresnel number, the flatness order of FTBs, the truncation of the aperture, as well as the relative incident distance. The results indicate that the focal shift away from the geometrical focus of the focused FTBs appears when the relative incident distance does not equal to 1 and increases with the decreasing of the Fresnel number or the beam flatness order for both the apertured and unapertured lens systems. In the lens system with aperture, the focal shift effect decreases with the increase of the truncation parameter.

Analysis of imaging properties of a microlens based on the method for a dyadic Green's function

The dyadic Green's function (DGF) is applied to examine the effect of focal shift in a spherical microlens with the variation of the numerical aperture for a given Fresnel number when a monochromatic plane wave with x linear polarization is incident on the microlens. By comparing the results based on the method for the vector Kirchhoff diffraction theory [J. Opt. Soc. Am. A22, 68-76 (2005)], the effect of the spherical aberration on focal shift in a microlens is evaluated, and the influences of NA as well as the spherical aberration on the transverse electric energy density distribution in the focal plane are also investigated. In contrast with other vector formulations of imaging theory, which mainly focus on the focal shift in an aplanatic system, the DGF method is more practical and effective to locate the principal maximum energy density along the normal axis and to study transverse electric energy density distribution, because the actual shape of a microlens and the effects of aberrations are considered.

Predictions of Rayleigh's diffraction theory for the effect of focal shift in high-aperture systems

In their work on diffraction [J. Opt. Soc. Am.51, 1050 1961], Osterberg and Smith have computed in an exact manner from the Rayleigh-Sommerfeld diffraction integral of the first kind the irradiance distribution along the axis of a converging spherical wave, and they found that in a scalar optical system of high relative aperture and finite value of Fresnel number, the central peak value of the axial irradiance may occur inside, at, or outside the geometrical focal point as the angular semiaperture of the system is less than, equal to, or greater than, respectively, a particular angle that falls near 70 degrees . These findings are now reexamined using a different assumption that takes into account diffraction at the edge of the aperture. Different results are obtained that agree well with the predictions of other theories of diffraction of light and give confidence to the common conclusions drawn by investigators of the effect of focal shift, that the point of the principal maximum of axial irradiance is not at the geometrical focus but shifted toward the aperture in systems of different relative aperture and finite value of Fresnel number.

Nonlinear focal shift in focused ultrasonic transducers and its dependence on the Fresnel number

In this work we present a systematic study of the on-axis location of the maximum pressure and intensity points in the field radiated by a focused transducer. We report the motion of these characteristic points as the transducer voltage is increased. Different initial distributions, ranging from uniform to Gaussian cases, are considered. The numerical analysis is based on the solutions of the KZK equation with different initial conditions. Experimental results have been obtained for three different Fresnel number focused transducers in linear and nonlinear regime in order to analyze the focal shift effect. An analytical expression of the initial (linear) shift of the maximum pressure position is given. This expression, and the results of the numerical simulations, shows a good agreement with the experimental data. As a main conclusion, we demonstrate that the axial range of the nonlinear shift of pressure is larger for strong initial focal shifts, occurring for small Fresnel number transducers. Theoretical and numerical predictions of the focal shift effect in Gaussian beams are also presented. In this work we have established the relation between the focal shift (both in linear and nonlinear regime) with the Fresnel number of the transducer.

Nonlinear change of on-axis pressure and intensity maxima positions and its relation with the linear focal shift effect

A comprehensive experimental, analytical and numerical study of the true focal region drift relative to the geometrical focus (focal shift effect) in acoustic focused beams and its nonlinear evolution is presented. For this aim, the concept of Fresnel number, proportional to the linear gain, is introduced as a convenient parameter for characterizing focused sources. It is shown that the magnitude of the shift is strongly dependent on the Fresnel number of the source, being larger for weakly focused systems where a large initial shift occurs. Analytical expressions for axial pressure distributions in linear regime are presented for the general case of truncated Gaussian beams. The main new contribution of this work is the examination of the connection between the linear and nonlinear stages of the focal shift effect, and its use for the estimation of the more complicated nonlinear stage. Experiments were carried out using a continuous-wave ultrasonic beam in water, radiated by a focused source with nominal frequency f = 1 MHz, aperture radius a = 1.5 cm and geometrical focal distance R = 11.7 cm, corresponding to a Fresnel number N F = 1.28. The maximum measured shifts for peak pressure and intensity were 4.4 and 1.1 cm, respectively. The evolution of the different maxima with the source amplitude, and the disparity in their axial positions, is interpreted in terms of the dynamics of the nonlinear distortion process. Analytical results for the particular case of a sound beam with initial Gaussian distribution are also presented, demonstrating that the motion of peak pressure and peak intensity may occur in opposite directions.

Focal shift and focal switch of focused truncated elliptical Gaussian beams

It is shown that when a generalized astigmatic elliptical Gaussian beam (EGB) propagates in a focusing optical system with an elliptical aperture, the effect of focal shift and focal switch can appear. An analytical propagation expression for an EGB through an axially nonsymmetrical optical system with a thin lens and an elliptical aperture is derived. Some numerical simulations are illustrated for its focal shift and focal switch, and some physical explanations are also given.

Focal shifts in diffracted converging electromagnetic waves I Kirchhoff theory

Starting with the vector formulation of the Kirchhoff diffraction theory, expressions for the total energy density distribution along the axis are presented without using any of the usual assumptions except the assumption made by Kirchhoff for the boundary conditions of a black screen. To make the Kirchhoff integral compatible with Maxwell's equations, a line integral around the edge of the aperture is added in the analysis. The consequence of ignoring the contribution of this line integral to the axial field distribution is examined numerically. The focal shift effect is investigated for both aplanatic systems and parabolic mirrors having an arbitrary numerical aperture (NA) and finite value of the Fresnel number. The combined effects of the Fresnel number and NA on the focal shift are evaluated, and the validity of the results is carefully checked by comparing the wavelength with the system dimensions.

Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number.

Analytic expressions are given for the on-axis intensity predicted by the Rayleigh-Sommerfeld and Kirchhoff diffraction integrals for a scalar optical system of high numerical aperture and finite value of Fresnel number. A definition of the axial optical coordinate is introduced that is valid for finite values of Fresnel number, for high-aperture systems, and for observation points distant from the focus. The focal shift effect is reexamined. For the case when the focal shift is small, explicit expressions are given for the focal shift and the axial peak in intensity.

The focal shift in converging waves

The focal shift effect in converging spherical waves diffracted by circular apertures is analysed, based on a geometrical interpretation of the mathematical expressions taken from scalar wave theory. The approach leads to a quasi-exact formula for the axial complex amplitude without the need for calculating any integral. The procedures can naturally be extended to image-forming instruments and an application to a single-lens system is also presented.

Three-dimensional behavior of apodized nontelecentric focusing systems.

The scalar field in the focal volume of nontelecentric apodized focusing systems cannot be accurately described by the Debye integral representation. By use of the Fresnel-Kirchhoff diffraction formula it is found that, if the aperture stop is axially displaced, the focal-volume structure is tuned. We analyze the influence of the apodizing function and find that, whereas axially superresolving pupil filters are highly sensitive to the focal-volume reshaping effect, axially apodizing filters are more inclined to the focal-shift effect.

Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system

The Debye formulation of focused fields has been systematically used to evaluate, for example, the point-spread function of an optical imaging system. According to this approximation, the focal wave field exhibits some symmetries about the geometrical focus. However, certain discrepancies arise when the Fresnel number, as viewed from focus, is close to unity. In that case, we should use the Kirchhoff formulation to evaluate accurately the three-dimensional amplitude distribution of the field in the focal region. We make some important remarks regarding both diffraction theories. In the end we demonstrate that, in the paraxial regime, given a defocused transverse pattern in the Debye approximation, it is possible to find a similar pattern but magnified and situated at another plane within the Kirchhoff theory. Moreover, we may evaluate this correspondence as the action of a virtual thin lens located at the focal plane and whose focus is situated at the axial point of the aperture plane. As a result, we give a geometrical interpretation of the focal-shift effect and present a brief comment on the problem of the best-focus location.