Polychromatic Optical Image. Diffraction Limited System and Influence of the Longitudinal Chromatic Aberration

The illuminance and the chromaticity distributions in the polychromatic diffraction pattern of an object point, formed by an optical system, are computed and discussed in this paper. We consider the aberration-free system, and systems with the most frequent states of longitudinal chromatic aberrations, i.e. apochromatism, achromatism and systems with lineal residual chromatic aberration. The study is carried out by determining in each case the colour (illuminance and chromaticity) in different image planes. The colour distributions in the diffraction patterns give more information about the aberration corrections than the study of only the illuminance distributions. The influence of the spectral emission is also studied by using the standard sources EQ (equal energy), A, B and C (1931 CIE). The standard observer (CIE 1931) is taken as the receiver.     

Calculation of retinal image quality for polychromatic light

Although the retinal image is typically polychromatic, few studies have examined polychromatic image quality in the human eye. We begin with a conceptual framework including the formulation of a psychophysical linking hypothesis that underlies the utility of image quality metrics based on the polychromatic point-spread function. We then outline strategies for computing polychromatic point-spread functions of the eye when monochromatic aberrations are known for only a single wavelength. Implementation problems and solutions for this strategy are described. Polychromatic image quality is largely unaffected by wavelength-dependent diffraction and higher-order chromatic aberration. However, accuracy is found to depend critically upon spectral sampling. Using typical aberrations from the Indiana Aberration Study, we assessed through-focus image quality for model eyes with and without chromatic aberrations using a polychromatic metric called the visual Strehl ratio. In the presence of typical levels of monochromatic aberrations, the effect of longitudinal chromatic aberration is greatly reduced. The effect of typical levels of transverse chromatic aberration is virtually eliminated in the presence of longitudinal chromatic aberration and monochromatic aberrations. Clinical value and limitations of the method are discussed.     

Hubble Space Telescope Faint Object Camera calculated point-spread functions

A set of observed noisy Hubble Space Telescope Faint Object Camera point-spread functions is used to recover the combined Hubble and Faint Object Camera wave-front error. The low-spatial-frequency wave-front error is parameterized in terms of a set of 32 annular Zernike polynomials. The midlevel and higher spatial frequencies are parameterized in terms of set of 891 polar-Fourier polynomials. The parameterized wave-front error is used to generate accurate calculated point-spread functions, both pre- and post-COSTAR (corrective optics space telescope axial replacement), suitable for image restoration at arbitrary wavelengths. We describe the phase-retrieval-based recovery process and the phase parameterization. Resultant calculated precorrection and postcorrection point-spread functions are shown along with an estimate of both pre- and post-COSTAR spherical aberration.     

Aberrations of Plane Grating Monochromators with Off-axis Paraboloidal Mirrors

Aberrations of plane grating monochromators with off-axis paraboloidal mirrors are studied analytically by the use of the light-path function and the Burch plate diagram extended to off-axis systems. Residual wave aberration of the third order becomes vanishingly small by compensation of two aberration terms in the U configuration, but remains fairly large by accumulation in the Z configuration. This explains an excellent image quality of the U configuration. It is shown that exact slit-image-curvature invariance with wavelength exists in the Z configuration but does not in the U configuration. A suitable value of the distance between the grating and off-axis paraboloidal mirror can be determined to eliminate certain residual aberration terms of the fourth order.     

Imaging of Extended Objects Using an Off-axis Holographic Lens

Abstract

The aim of this paper is to evaluate the performance of an off-axis holographic lens afflicted with third-order spherical aberration and coma using a numerical ray-tracing procedure. The light intensity distribution in the aberration spot observed at the Gaussian image plane has been evaluated for different amounts of coma and for a fixed value of spherical aberration. The images of both an extended coherent and an incoherent edge as well as bar objects have been computed and the results compared with the aberration-free case. All the results are illustrated in graphical form.     

Imaging of Extended Objects Using a Curved Holographic Lens Free From Spherical Aberration

Abstract

This paper evaluates the performance of an off-axis curved holographic lens using a numerical ray-tracing procedure. The hologram curvature has been suitably selected to eliminate spherical aberration. Light intensity distribution in the aberration spot observed at the Gaussian image plane has been evaluated for different object fields. The images of extended coherent and incoherent edge as well as bar objects have been computed and the results compared with that for the aberration-free case. All the results are illustrated in graphical form.     

Aberrated Optical Systems with Optimum Apodizers for the Strehl Ratio: Axial and Extra-axial Point-spread Function and Modulation-transfer Function

Abstract

The amplitude distribution at the aperture of an optical system for off-axis object points due to the introduction of a non-uniform transmission filter depends on the transmission function and on the longitudinal filter position. The effects of several filters on the axial and extra-axial point-spread function (PSF) and modulation-transfer function (MTF) of an optical system for different filter positions are studied. The filter transmission function and the filter position are obtained by the optimization of the axial and extra-axial Strehl ratios. The axial and extra-axial effects of the filters on the PSF and on the MTF are analogous. There are some positions of the filter for which the overall performance is significantly improved. The optimum longitudinal positioning of the apodizing filter is achieved in a process which is analogous to the stop shift effect in conventional optical design. The lack of radial symmetry in the off-axis wavefront aberration function at any particular field angle is convolved with the apodizing function and this leads to the anticipated improvement.     

Exact ray-trace beam for an off-axis paraboloid surface

When an off-axis paraboloidal mirror focuses a parallel beam, the image is formed on one side of the optical axis. For a tilted beam focused by an off-axis paraboloidal mirror, the focus is no longer pointlike (not considering the diffraction effect); rather, it is a distorted spot. This is due to the inherent aberrations of the surface. In addition, there is a change in the focus position. We calculate by exact ray-trace equations the modified wave-front aberration and express it in power series. Our formulation uses the optical path variation along a defined principal ray that we relate to the parameter that describe the surface and the beam angle of incidence. We designate this ray as that reflected by the center of the entrance pupil and field of view. We employ the direction cosines of the principal ray to compute the wave-front aberration function of a beam reflected by an off-axis paraboloid.     

Geometrical theory of aberrations near the axis in classical off-axis reflecting telescopes

A geometrical theory of aberrations for the vicinity of the focus of arbitrary off-axis sections of conic mirrors is derived. It is shown that an off-axis conic mirror introduces linear astigmatism in the image. However, in classical two-mirror telescopes this aberration can be eliminated by tilting the secondary parent mirror axis. It is also shown that the practical geometrical-optics performance of a classical off-axis two-mirror telescope with no linear astigmatism is equivalent to the performance of an on-axis system, proving that both systems have identical third-order coma. To demonstrate the applicability of the theory developed in a practical system, a fast (i.e., f/2), compact, obstruction-free classical off-axis Cassegrain telescope is designed.     

Effects of index-mismatch-induced spherical aberration in pump--probe microscopic image formation

Pump--probe fluorescence microscopy has been demonstrated to be a powerful tool for obtaining three-dimensional, time-resolved information in bioimaging applications. However, the use of this technique can be complicated by the fact that the different wavelengths used to achieve pump--probe microscopy can result in wavelength-dependent spherical aberration, thus limiting the usefulness of the technique. We address this issue by investigating the effects of refractive-index-mismatch-induced spherical aberration on pump--probe image formation. We model the effects by considering pump--probe imaging performed with an objective with a numerical aperture of 0.75 focusing through an oil-water interface. Our results show that spherical aberration has the greatest effect in degrading an axial point-spread function. In addition to signal loss, the redistribution of signal strength along the axial direction results in broadening of the FWHM of the plane response function. The inclusion of confocal detection tends to improve image resolution but at a significant loss of signal strength.     

Effect of a Low Efficient Shutter on Balanced Aberrations

The effect of a 50 per cent efficient between-the-lens shutter which may be considered as one of the typical low efficient shutters is studied from the viewpoint of balanced aberrations. A low efficient between-the-lens shutter which operates circularly produces an effect similar to a shaded aperture in which the amplitude transmittance decreases from the centre of the aperture to its rim. The results of balanced aberrations calculated from the Strehl definition of an exposure image of the point source are compared with both the clear aperture and the shaded aperture. It is shown that the balanced aberrations are affected by a 50 per cent efficient shutter. When the aberration has only primary spherical aberration or primary coma, the shift of best focal setting caused by the aberration is reduced by using a 50 per cent efficient shutter.     

Fresnel diffraction by a circular aperture with off-axis illumination and its use in deconvolution of microscope images

The Fresnel approximation for off-axis illumination of a circular aperture is reexamined. The point-spread function for an aberration-free system can be expressed in terms of redefined optical coordinates. An improved expression is given for contours of constant intensity in the focal plane. The variation in axial width of the focal spot with angle of offset is discussed. The predictions are compared with exact calculations of the Rayleigh-Sommerfeld diffraction integral. Limitations for application in deconvolution of microscope images formed with objectives of finite tube length are discussed.     

Diversity selection for phase-diverse phase retrieval

Wavefront-sensing performance is assessed for focus-diverse phase retrieval as the aberration spatial frequency and the diversity defocus are varied. The analysis includes analytical predictions for optimal diversity values corresponding to the recovery of a dominant spatial-frequency component in the pupil. The calculation is shown to be consistent with the Cramér-Rao lower bound by considering a sensitivity analysis of the point-spread function to the spatial frequency being estimated. A maximum value of diversity defocus is also calculated, beyond which wavefront-sensing performance decreases as diversity defocus is increased. The results are shown to be consistent with the Talbot imaging phenomena, explaining multiple periodic regions of maximum and minimum contrast as a function of aberration spatial frequency and defocus. Wavefront-sensing performance for an iterative-transform phase-retrieval algorithm is also considered as diversity defocus and aberration spatial frequency are varied.     

Design of a compact off-axis two-mirror freeform infrared imager with a wide field of view

With the development of modern precision optical fabrication and measurement technologies, optical freeform surfaces have been widely employed in different applications, especially in off-axis reflective optical systems. For an infrared optical system operating in the long-wavelength spectrum, compactness, brightness, and a wide field of view are key requirements for military surveillance or scene sensing. In this paper, we present an off-axis two-mirror freeform infrared imager with compactness and brightness. It has a large pupil of size 12?mm and a fast focal ratio of 2.2 over a wide 23° diagonal field of view, as well as good image quality. XY-polynomial freeform surfaces are applied to the viewing mirror and focusing mirror. The multiple degrees of freedom of optical freeform surfaces are very helpful for off-axis aberration correction and improving the optical performance over the entire pupil across the full field of view. The overall dimension of our designed freeform infrared imager is about 30?mm by 30?mm by 30?mm, which is elegantly miniaturized. The final designed results of a reflective freeform infrared imager are demonstrated and discussed in detail.     

Varied line-space grating for flat spectral response of coupling to single-mode fiber

We use a planar linear grating with varied line-space grooves to introduce a tailored one-dimensional phase variation profile that results in an aberrated point-spread function at the focal plane. A design procedure for the period chirp map for such gratings is developed. As an example, we present theoretical and experimental results on a mechanically ruled, varied line-space echelle grating in single-mode fiber-coupled optical multiplexers in the wavelength region of 1545 nm. The varied line-space grating changes the multiplexer's Gaussian spectral response function to a flat-top dependence with reduced sensitivity to source laser wavelength drift.     

Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions

We assess the validity of an extended Nijboer-Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on ecently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.     

The Computation of Diffraction Patterns in the Presence of Aberrations

The diffraction integral for the disturbance produced in the image plane normal to the optical axis by an extra-axial pencil has been shown to lead to a Fourier transform provided the exit pupil surface is taken to be that of the reference sphere. It has been shown also that, except for small aperture and field sizes, the effect on the wave-front aberration of a shift of the image plane is not represented with sufficient accuracy merely by a term proportional to the aperture squared. Both of these results have been respected in formulating a numerical technique for the calculation of point spread functions. The diffraction integral is evaluated in polar coordinates, and is such that no error is made in approximating the domain of the exit pupil in cases where this may be represented by an ellipse. A study of the accuracy obtained has shown that, if each quadrant of the pupil is divided into a 20 × 20 mesh of elementary areas, the error in the intensity is not expected to exceed 0·8 per cent of the intensity at the focus of a diffraction limited system. The method takes account of the first derivatives of the wave aberration at each mesh point, and the results are therefore expected to be more accurate than those obtained by merely replacing the integral by a simple sum. Results are given for a case of primary and secondary coma, and of a study of the influence of secondary spectrum and spherical aberration on the images formed by 2 mm achromatic microscope objectives of numerical aperture equal to 1·40.     

陶瓷标准色板的研制

介绍也以色度学三原色为依据研制的陶瓷色度板，它由景德镇出产的彩色釉面瓷块经过特殊烧结而成，具有白、红、绿、蓝和黄五种颜色。其表面平整，颜色均匀，色饱和度高，光学性能和物化性能稳定，覆盖了整个可见光谱波长：可作为颜色测量用的标准计量器具。     

First-generation holographic, grazing-incidence gratings for use in converging, extreme-ultraviolet light beams

We present two holographic recording solutions that produce gratings suitable for use at grazing incidence in the extreme ultraviolet. The rulings are formed when the interference pattern of two spherical wave fronts is recorded on a planar substrate. Each grating is designed to minimize or eliminate the dominant aberration terms in order to maximize the spectral and spatial resolution of the system. In the first design, the dominant astigmatism term in a power-series expansion of the light path function is eliminated; in the second design, the dominant comatic terms are minimized. Each grating is placed directly in a converging light beam at grazing incidence to provide high system efficiency in the extreme ultraviolet. The aberration control afforded by both recording solutions is excellent, providing detector-limited spatial and spectral resolution over much of the usable bandpass. Furthermore, the aberration control is maintained over a wide range of beam speeds and off-axis angles, thereby outperforming conventional varied line-space gratings for use in the extreme ultraviolet. We discuss the methodology used to develop the recording solutions, model and compare the performance of the gratings, and discuss possible space-based applications for these gratings.     

Influence of intraocular scattered light on lightness-scaling experiments

Following Munsell's bisection procedure [J. Opt. Soc. Am. 23, 394 (1933)], we established a nine-step gray scale in which each step is an equal increment in lightness. We calculated retinal illuminances after intraocular scatter by using the point-spread function of Vos et al. [Vision Res. 16, 215-219 (1976)]. After this correction for intraocular scatter, we find a logarithmic relationship between retinal illuminance and achromatic lightness scales that are determined by the bisection method. Additional bisection experiments with a series of different backgrounds corroborate this result. We find that lightness depends linearly on the logarithm of scatter-corrected retinal illuminance, with different slopes for backgrounds of different lightness. This study also highlights the importance of using scatter-corrected illuminance in any quantitative model of lightness.