Low Fresnel Number Research Articles

Refined beam expansion method for unstable resonator modes

A refined and simplified version of the beam expansion method for computing the modes of unstable resonators is presented here.The principle of the beam expansion method is to express the field pattern on a basis of Hermite-Gaussian functions. Propagating each individual component around the resonator enables a transfer matrix to be constructed from which the eigenvalues and eigenvectors of the modes can be determined.Whereas the earlier proof-of-principle approach performed well for low Fresnel numbers, the refined technique gives good results for the challenging test case of magnification M=1.9 and equivalent Fresnel number Neq=49.4. The performance enhancement has been achieved by incorporating a number of adjustments that improve the accuracy of the calculation.

Achromatic flat lens performance limits

In recent years there has been a lot of interest in flat lenses, a category that includes diffractive lenses and metalenses. These lenses have the potential of reducing the size and cost of optical systems by replacing conventional refractive optical elements. A major obstacle to the widespread use of flat lenses is their inherent large chromatic aberration, associated with diffraction effects. To meet this challenge, achromatic diffractive lenses and metalenses have been developed. In this paper we review previously reported metalens performance limits, derive limits on the performance of achromatic diffractive lenses, and compare the two. We show that such lenses can support a wide spectral range, limited only by loss of efficiency caused by manufacturing limitations related to feature depth and size. On the other hand, we show that these lenses can provide near-diffraction-limited performance only at very low Fresnel numbers, i.e., they cannot provide large focusing power and broadband response simultaneously. We then go on to compare the limits of achromatic metalenses and diffractive lenses, in attempt to understand the potential of different types of flat lenses. Our findings facilitate better understanding of flat lens capabilities and limitations, and the exploration of novel design concepts and applications.

Accounting for focal shift in the Shack-Hartmann wavefront sensor.

The Shack–Hartmann wavefront sensor samples a beam of light using an array of lenslets, each of which creates an image onto a pixelated sensor. These images translate from their nominal position by a distance proportional to the average wavefront slope over the corresponding lenslet. This principle fails in partially and/or non-uniformly illuminated lenslets when the lenslet array is focused to maximize peak intensity, leading to image centroid bias. Here, we show that this bias is due to the low Fresnel number of the lenslets, which shifts the diffraction focus away from the geometrical focus. We then demonstrate how the geometrical focus can be empirically found by minimizing the bias in partially illuminated lenslets.

Optimizing three-dimensional point spread function in lensless holographic microscopy

In a number of previous studies on light focusing, the asymmetric axial intensity distribution with intensity peak shifted away from the paraxial focal plane was demonstrated for lenses working with a low Fresnel number. Here, the axial asymmetry of the three-dimensional point spread function (PSF) and the aberration effects are examined in a magnified phase-shifting holographic imaging achieved by the mismatch of reference and reconstruction waves. In the analysis, an optimal combination of experimental parameters and the range of applicable lateral magnifications are found for which the axial asymmetry of the PSF is not apparent and the aberration effects are acceptable. The focal shift and the axial asymmetry of the PSF and the effects of holographic aberrations are evaluated by approximate quantitative criteria whose validity is verified in exact numerical models and experiments. The optimal design of in-line holographic geometry is demonstrated by reconstructing the three-dimensional PSFs and the image of the resolution target recorded in the experimental setup using a spatial light modulator.

On the Nonlinear Effects in Focused Ultrasound Beams with Frequency Power Law Attenuation

When finite amplitude ultrasound propagation is considered, changes in spatial features of focused ultrasound beams can be observed. These nonlinear effects typically appear in thermoviscous fluids as focal displacements, beam-width variations or gain changes. However, in soft-tissue media, the frequency dependence of the attenuation doesn’t obey a squared law. In this way, these complex media response leads to weak dispersion that prevents the cumulative processes of energy transfer to higher harmonics. In this work we explore the influence of different frequency power law attenuation responses and its influence on the self-defocusing effects in focused ultrasound beams. Thus, we numerically explore the spatial field distributions produced by low-Fresnel number devices and High Intensity Focused Ultrasound (HIFU) radiating trough different soft-tissue media.

Phase retrieval applied to stellar occultation for asteroid silhouette characterization.

Here we expand on the current methods of characterizing small astronomical bodies, particularly asteroids, by viewing stellar occultation events. Stellar occultation has proven to be a viable method for determining the size of moons and asteroids; however, it comes with some limitations. In general the method does not consider or use all of the known diffraction effects that occur and thus provides a nominal radius--not a shape--of the occluder. We show that most stellar occultation events involving small near-Earth asteroids occur with low Fresnel numbers. This in effect renders the traditional methods useless to characterize the shape, because no sharp shadow exists. We show that using similar data collection to that of the traditional occultation method and inverting a Fresnel diffraction equation by a phase retrieval process can yield a complete reconstruction of the silhouette of the occluder. The effect of noise in the measurements is also discussed. A practical example applied to the asteroid 25143 Itokawa is shown.

Focusing properties of aberration-free electromagnetic waves in thin dielectric slabs

We study properties of electromagnetic fields that are focused into thin dielectric slabs using an approach based on physical-optics or Kirchhoff’s approximation, making the results valid both for low and high Fresnel number focusing geometries. A variety of thin slab structures and focusing systems are studied with emphasis on the impact of the field that is due to the presence of the back interface of the slab which is critical in various fields such as optical recording, bio and material sciences. In particular, we investigate how the intensity of the focused field depends on variables such as aperture size, aperture-slab distance, slab refractive index, and slab thickness. Numerical results show that the contribution from the electric field due to the presence of the back interface of the slab is significant along the axis, which passes through the center of the spherical aperture and the focal point. With fixed focal length, the axial field inside the slab is found to increase and the focal shift to decrease as the aperture-slab distance is reduced. With increasing relative refractive index of the slab, the axial field is found to decrease, but to be more concentrated around the focus. With increasing aperture size, the axial field is found to increase, while interference effects due to the presence of the back interface of the slab are found to decrease.

Focusing of electromagnetic waves into a dielectric slab. II. Numerical results

Based on existing exact and asymptotic Kirchhoff solutions for focused electromagnetic fields inside a dielectric slab we present numerical comparisons between them also for the special cases of focusing through a single interface or in a single medium. These comparisons show that the exact and asymptotic Kirchhoff solutions for focusing in a single medium or through a single interface agree well, except at observation points near to the interface, while a small difference between the two solutions for the focused electrical field inside a dielectric slab is observed, especially at observation points near to one of the interfaces. This difference is believed to be due to contributions from surface waves, which are not accounted for in the asymptotic Kirchhoff solutions. At low Fresnel numbers focal shift phenomena are observed in all three cases.

Analysis of depth-of-focus of synthetic aperture antenna arrays

Interferometric synthetic aperture arrays (SAA) can be refocused at any range via digital processing of the raw data, called the visibility function. Such systems are sensitive, however, to ranging errors in digital refocusing. They, therefore, exhibit a limited depth of focus, because these errors are equivalent to introducing defocus in the system. We derive an analytical expression for the Strehl ratio of synthetic aperture arrays, which accounts for both the antenna voltage pattern and the focal shift factor encountered in imaging systems with a low Fresnel number. Assessment of the depth of focus of short-range imaging arrays is allowed and is illustrated for common array designs.

Design considerations for low-light level low-Fresnel number optical systems

Low-Fresnel number optical systems exhibit significant diffraction effects that cause a shift in the peaks of on-axis irradiance away from the geometric focal point. This is currently interpreted as a change of the focal length of an optical system, leading optical system designers to compensate for the effect by assuming the image plane is coincident with the peak of on-axis irradiance. While this may be an appropriate interpretation for certain applications, I show that despite the shift in peak irradiance away from the geometrical focal point, a change in a system's optical power will not increase the on-axis irradiance at that distance. This is important for low-light level applications where it is necessary to mitigate diffraction induced transmission losses. I also show that low-Fresnel number systems have increased tolerance on system power at the geometrical focal point and as a result are inherently achromatic.

Focal shifts on focusing through a plane interface

We derive explicit formulas for the focal shift, the intensity increase, and the axial zeros obtained on focusing at low Fresnel numbers through a plane interface between two isotropic media, and demonstrate their validity through comparisons with numerical plots of axial intensity distributions. Also, we show that for the special case in which the two isotropic media have the same refractive index, these formulas are in complete agreement with previous results presented for focusing in a single medium.

Super focusing of optical beams

It was commonly believed that the spherical wavefront aberration of an optical beam always leads to a lower maximum focused intensity along the optical axis. It was shown recently both theoretically and experimentally that this was only true for high Fresnel number systems. In low Fresnel number focusing systems a properly chosen spherical wavefront aberration can increase the maximum focused intensity. In this paper, we call this 'super focusing'. To investigate the characteristics of the super focusing, an improved axial intensity formula has been newly derived for Gaussian beams with spherical aberrations. Based on this formula, several examples of axial intensity distributions are shown. The cause of the super focusing is considered based on these intensity distributions and the derived formulae.

On-axis irradiance distribution of axicons illuminated by spherical wave

The focusing properties of an axicon illuminated by spherical wave are investigated. We define the Fresnel number of the focusing axicon impinged on by spherical wave. The on-axis irradiance distribution versus Fresnel number is studied. It is shown that, similar with the plane wave, focal squeeze, which is defined as deviations for the position of the maximum irradiance, also appears in low Fresnel number. Besides, the on-axis irradiance will be smoothed for appropriate curvature radius of the divergent spherical wave; meanwhile, when the convergent spherical wave is incident on the axicon, the on-axis irradiance varies more seriously.

Phase space models for stochastic nonlinear parabolic waves: wave spread and collapse

We derive several kinetic equations to model the large scale, low Fresnel number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly fluctuating random potential. There are three types of kinetic equations the longitudinal, the transverse and the longitudinal with friction. For these nonlinear kinetic equations we address two problems: the rate of dispersion and the singularity formation. For the problem of dispersion, we show that the kinetic equations of the longitudinal type produce the cubic-in-time law, that the transverse type produce the quadratic-in-time law and that the one with friction produces the linear-in-time law for the variance prior to any singularity. For the problem of singularity, we show that the singularity and blow-up conditions in the transverse case remain the same as those for the homogeneous NLS equation with critical or supercritical self-focusing nonlinearity, but they have changed in the longitudinal case and in the frictional case due to the evolution of the Hamiltonian.

Strong on-axis focal shift and its nonlinear variation in low-Fresnel-number ultrasound beams

We examine the case of focusing transducers with a strong initial (in linear regime) shift of the main on-axis peak pressure maximum from the geometrical focal point toward the transducer. Such transducers are characterized, using the concepts introduced in this paper, by a low Fresnel number. The displacement of this initially shifted on-axis pressure maximum point toward the geometrical focus, and its backward motion as the driving transducer voltage increases until highly nonlinear regimes, has been experimentally observed. The simultaneous monitoring of the temporal wave-form distortion determines the real roles and interplay between different nonlinear effects (refraction and attenuation) in the observed dynamics of the on-axis pressure maximum. The numerical solution of the corresponding mathematical model confirms the physical interpretation of the observed phenomenon.

On the chromatic aberration of microlenses

The optical properties of plano-convex refractive microlenses with low Fresnel Number (typically FN < 10) are investigated. It turns out that diffraction effects at the lens aperture limit the range of the effective focal length. The upper limit of the focal length is determined by the diffraction pattern of a pinhole with equal diameter. In addition achromatic microlenses can be realized because refraction and diffraction have opposing effects on the focal length. Gaussian beam propagation method has been used for simulation. The presented results are of relevance for applications, where microlenses with small apertures and long focal lengths are used, for example, Shack Hartmann wavefront sensors or confocal microscopes.

Axial irradiance of a focused beam

The principal maximum of axial irradiance of a focused beam with a low Fresnel number does not lie at its focal point; instead it lies at a point that is closer to the focusing pupil. It has been shown by the numerical example of a weakly truncated Gaussian beam that its value increases and its location moves closer to the pupil when spherical aberration is introduced into the beam. Such an increase has been referred to as "beyond the conventional diffraction limit." Similarly, an increase in the value and a shift in the location of the principal maximum of axial irradiance of a uniform beam toward the pupil by the introduction of some spherical aberration has been characterized as an unexpected result. We explain why and how such a result comes about and that it neither invalidates any diffraction limit nor is it unexpected. We illustrate this for uniform as well as Gaussian beams of various truncation ratios. Both focused and collimated beams aberrated by spherical aberration or astigmatism are considered.

Focal shifts of converging diffracted waves of any state of spatial coherence

We analyze the focusing of wave fields of any state of coherence by systems with low Fresnel numbers. We study the optical intensity on the axis in the focal region. The dependence of the focal shift and of the maximum on-axis intensity on the state of coherence is examined for some model fields.

Comparison of focusing properties of conventional and diffractive lenses

We compare focusing properties of conventional lenses and diffractive lenses (DLs). To that end, we use a converging electric-dipole wave as a model of an optimal converging wavefront that emerges from a conventional lens system. Our numerical results show that DLs have better focusing properties than conventional lenses for f-numbers less than 1. Although both types of lenses give nearly indistinguishable co-polarized intensity components in the focal region at low f-numbers, DLs provide much lower longitudinally polarized components. However, at sufficiently low Fresnel numbers and high f-numbers DLs fail to provide a well-focused energy distribution.

Excess noise in low Fresnel number unstable resonators

Numerical calculations of excess noise factors in low Fresnel number unstable resonators are described in detail. Computed mode profiles in one transverse dimension together with associated Petermann K-factors are presented; dynamical considerations such as injected wave excitation are also examined. The properties of the zero-order modes are consistent with virtual source theory and with a simple formula for K based on a geometrical optics approximation. While virtual source theory is asymptotic, we find that it can make good predictions for Fresnel numbers as low as unity. Full numerical calculations are however needed to determine accurate mode profiles and K-factors in some regimes. A new technique for calculating accurate higher-order mode profiles is also developed and this is employed to evaluate K-factor dependencies of the first two higher-order even modes.