Phase plate to extend the depth of field of incoherent hybrid imaging systems

A hybrid imaging system combines a modified optical imaging system and a digital postprocessing step. We describe a spatial-domain method for designing a pupil phase plate to extend the depth of field of an incoherent hybrid imaging system with a rectangular aperture. We use this method to obtain a pupil phase plate to extend the depth of field, which we refer to as a logarithmic phase plate. Introducing a logarithmic phase plate at the exit pupil of a simulated diffraction-limited system and digitally processing the detector's output extend the depth of field by an order of magnitude more than the Hopkins defocus criterion. We also examine the effect of using a charge-coupled device optical detector, instead of an ideal optical detector, on the extension of the depth of field. Finally, we compare the performance of the logarithmic phase plate with that of a cubic phase plate in extending the depth of field of a hybrid imaging system with a rectangular aperture.     

Comment on ‘extended depth of field in hybrid imaging systems: circular aperture’

A recent paper by Sherif et al. (S.S. Sherif, E.R. Dowski, W.T. Cathey, J. Mod. Opt. 51 1191 (2004).) reported the derivation of an aspherical phase plate, which when placed at the exit pupil of a conventional imaging system and combined with digital processing of the recorded images, increases the depth of field by an order of magnitude. An error in the derivation of this phase plate has been identified, which makes the reported extension of depth-of-field sub-optimum rather than invalid. In this comment, an optimum phase plate is obtained and the relevant results are repeated.     

大景深光学成像系统关键技术的研究

在普通光学成像系统中引入相位模板和数字图像复原技术，可以增大系统的景深。本文设计了一种新的由多个等面积环形区域组成的纯相位模板，相邻的环形分区交替补偿高斯物面两侧离焦。同时，基于频域滤波理论和MTF离焦不变性质，给出了相应的不依赖于离焦程度的图像复原滤波器的表达式。系统的MTF曲线、MTF密度图和Spoke图结果表明系统具有大景深特性。通过适当调整模板参量，系统景深可以达到普通光学系统的4倍以上。系统可以应用到生物和医学的显微成像、机器视觉等多个领域。     

Reduced depth of field in incoherent hybrid imaging systems

A hybrid imaging system combines a modified optical imaging module and a digital postprocessing step. We define what to our knowledge is a new metric to quantify the blurring of a defocused image that is more suitable than the defocus parameter for describing defocused hybrid imaging systems. We use this metric to design a pupil phase grating to reduce the depth of field, thereby increasing the axial resolution, of an incoherent hybrid imaging system using quasi-monochromatic illumination. By introducing this grating at the exit pupil and digitally processing the output of the detector, we reduce the depth of field by more than a factor of 2. Finally, we examine the effect of using a CCD optical detector, instead of an ideal optical detector, on the reduction of the depth of field.     

Extended depth of field with the optimal ‘0, π’ binary phase pupil mask

A ‘0, π’ phase pupil mask was developed to extend the depth of the field of a circularly symmetric optical imaging system. A global search algorithm was used to seek an optimal pupil mask which provides the largest spatial frequency band in a certain desired contrast value. The modulation transfer function curves and the normalized point spread function figures of the imaging system with the optimal mask were analyzed. The results show that the imaging system has a high resolution in a long frequency band and can obtain clear images without any post-processing. The experimental results also demonstrate that the depth of field of the imaging system is extended sixfold successfully.     

Realizations of focus invariance in optical-digital systems with wave-front coding

We report experimental verification of an extended depth of focus (EDF) system with near-diffraction-limited performance capabilities. Dowski and Cathey [Appl. Opt. 34, 1859-1866 (1995)] described the theory of this system in detail. We can create an EDF system by modifying a standard incoherent optical system with a special cubic phase plate placed at the aperture stop. We briefly review the theory and present the first optical experimental verification of this EDF system. The phase plate codes the wave front, producing a modified optical transfer function. Once the image is transformed into digital form, a signal-processing step decodes the image and produces the final in-focus image. We have produced a number of images from various optical systems using the phase plate, thus demonstrating the success of this EDF system.     

Imaging with Fresnel zone pupil masks: extended depth of field

Fresnel zone plates are used as pupil apertures to extend the depth of field of an incoherent imaging system. The defocused optical transfer function is evaluated approximately, and the results are confirmed by experiments and by subjective evaluations of the images of a test chart. With a pupil having four zones, for example, the resolution limit at four depths of field out of focus is improved by a factor of 10.     

Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function

In this paper we derive an approximate analytical representation for the modulation transfer function (MTF) of an imaging system possessing a defocused cubic-phase pupil function. This expression is based on an approximation using the Arctan function and significantly reduces the computational time required to calculate the resulting MTF. We derive rigorous bounds on the minimum and average accuracy of our approximation. Using this approximate representation of the MTF, the analytical solution of the problem of calculating the extension of the depth of field for a circular aperture with a cubic phase mask becomes possible. We also comment on how one can modify our method to construct a lower-bound or an upper-bound approximate analytical expression for the MTF.     

Quality parameters analysis of optical imaging systems with enhanced focal depth using the Wigner distribution function

An analysis of the Strehl ratio and the optical transfer function as imaging quality parameters of optical elements with enhanced focal length is carried out by employing the Wigner distribution function. To this end, we use four different pupil functions: a full circular aperture, a hyper-Gaussian aperture, a quartic phase plate, and a logarithmic phase mask. A comparison is performed between the quality parameters and test images formed by these pupil functions at different defocus distances.     

Focal shift and extended focal depth with tunable pupil filter

Extended focal depth and focal shift are very important in microscopy, imaging and optical storage systems, and have attracted much attention in recent years. In order to obtain the extended focal depth and focal shift, a new kind of tunable pupil filter is proposed in this article. It consists of one half-wave plate between two quarter-wave plates, and the half-wave plate is made up of two zones that can rotate with respect to each other. By analyzing the intensity distribution in the focal region of the optical system with such a device, it reveals that focal shift can be realized by rotating any zone of the half-wave plate. When the phase difference of the two zones is π, the extended focal depth and transverse superresolution can be obtained at the same time. Therefore, it may be feasible to use such a tunable pupil filter in optical systems that need focal shift and extended focal depth.     

Imaging Properties of the Schwarzschild Soft-X-ray Microscopes

Abstract

The imaging properties of the soft-X-ray microscopes composed of a Schwarzschild objective and a grazing incidence mirror condenser are discussed. Fundamental formulae on the partially coherent transfer function, the resolution of a two-point object and the edge response have been derived by consideration of the optical properties of an object in soft-X-ray regions. Then these imaging properties for the microscopes are investigated with the formulae and compared with those for a normal circular pupil system.     

波前编码系统中相位掩膜板的新设计

波前编码技术能扩大非相干光学成像系统的景深。根据传统相位掩膜板的设计原理,从两个方面对原有的相位掩膜板农达式进行扩展。由此提出了两种新的设计方案：“多项式”方案和“高次方”方案。前者是在原有的公式中加入了x平方项,而后者是将原有的高次方指数拓展到小数形式。将这两种新系统进行模拟计算,得到了相关的调制传递函数（MTF）数据和三维图,结果显示MTF随离焦量变化很小（当u=1时,均方差分别为0．0027和0．0021）,而且不会出现零值,跟传统方案MTF差值分别在0．05和0．04以内。由此说明新方案也能实现扩大光学系统景深的目的。     

Simplified EDF characterization method for WDM applications

Abstract

A new, simple EDF characterization method is presented. Departing from well-known equations, some approximations, applicable in WDM working conditions, are introduced. Thanks to them, the number of active parameters necessary to describe the EDF behaviour is sensibly reduced. Afterwards, a simple experiment to find such parameters is proposed. Finally, the characterization method is applied to an EDF and the parameters′ values obtained are used to predict gain results. A good agreement between these predictions and experimental data is observed.     

Comparison of depth-of-focus-enhancing pupil masks based on a signal-to-noise-ratio criterion after deconvolution

We consider optimization of hybrid imaging systems including a pupil mask for enhancing the depth of field and a digital deconvolution step. In a previous paper [Opt. Lett. 34, 2970 (2009)] we proposed an optimization criterion based on the signal-to-noise ratio of the restored image. We use this criterion in order to optimize different families of phase or amplitude masks and to compare them, on an objective basis, for different desired defocus ranges. We show that increasing the number of parameters of the masks allows one to obtain better performance.     

Application of bispectral speckle imaging to near-diffraction-limited imaging in the presence of unknown aberrations

A laboratory experiment that demonstrates near-diffraction-limited imaging of a detailed object in the presence of unknown fixed aberrations in the imaging system is described. A random-phase plate is introduced in a pupil plane of the imaging system to eliminate the effect of fixed aberrations in the system. We employ a bispectral speckle imaging technique to recover the object from speckled images affected by both the random-phase fluctuations induced by the random-phase plate and the fixed aberrations present in the imaging system. For the case where the random phase is assumed to obey Gaussian statistics an approximate form of the bispectral speckle transfer function is obtained with an asymptotic expansion. This approximate form of the transfer function shows the diffraction-limited nature of bispectral speckle imaging when the standard deviation of the random-phase fluctuations is of the order of a wavelength of light. Experimental results are presented for fixed aberrations associated with lens tilt and defocus in the imaging system.     

Pattern Recognition and Image Preprocessing

Abstract

In this paper a numerical algorithm for calculation of the point spread function from the complex light amplitude known on the output pupil of the imaging system is described. The first step of this algorithm includes a global polynomial approximation of the input data given in a number of points distributed arbitrarily over the entire output pupil. In the next step the numerical calculation of the diffraction integral is performed. After transition to polar coordinates and dividing the region of integration into a number of infinitesimal subregions of square shape the exponent in the integrand is approximated locally by a polynomial and then the whole integrand is approximated by a polynomial multiplied by the exponential function with linear exponent. Finally the respective integration is performed analytically. The analysis of the accuracy obtained as well as the results of exemplary calculations prove that the described algorithm is very useful in the analysis of image quality of such imaging elements as hybrid (diffractive–refractive) lenses.     

Calculation of vectorial three-dimensional transfer functions in large-angle focusing systems

The optical transfer function (OTF) is used in describing imaging systems in the Fourier domain. So far the calculation of the OTF of a large-aperture imaging system has been difficult because the vectorial nature of light breaks the cylindrical symmetry of the pupil function. We derive a simple line integral solution for calculating the vectorial three-dimensional OTF. We further extend this approach to imaging through a planar interface of two media with mismatched refractive indices. In general, our formalism allows for calculation of the Fourier transform of any product of two arbitrary vector components of the electromagnetic field. Arbitrary phase or amplitude modifications of the pupil function can be taken into account.     

光瞳滤波提高投影光刻成像分辨力研究

针对投影光刻成像系统在数值孔径足够大时所产生的分辨力和焦深的矛盾,详细研究了光瞳滤波对投影成像对比度的改善情况,根据不同掩模图形设计对应的最优滤波器。研究结果表明,光瞳滤波能大幅度提高投影光刻成像分辨力并增大焦深,是一种比较有效的提高光刻成像分辨力的方法。     

Numerical aspects of wave forces on an artificial island

Abstract

The wave forces on an artificial island standing on the sea‐bed are investigated by the proposed hybrid method. A boundary integral equation is developed for the inner fluid domain encompassing the arbitrarily shaped island, while an eigenfunction expansion is used to represent a simple velocity field in the outer fluid domain. Applying the natural boundary condition on the matching surface, the wave force and moment on the wetted surface of the island are found according to the calculated velocity potentials. A gravity step circular dock is used as a test case to confirm the present computation. As a practical application of the present method, numerical results for an elliptical artificial island are presented.