Frequency analysis of the wavefront-coding odd-symmetric quadratic phase mask

A mathematical analysis of the frequency response of the wavefront-coding odd-symmetric quadratic phase mask is presented. An exact solution for the optical transfer function of a wavefront-coding imager using this type of mask is derived from first principles, whose result applies over all misfocus values. The misfocus-dependent spatial filtering property of this imager is described. The available spatial frequency bandwidth for a given misfocus condition is quantified. A special imaging condition that yields an increased dynamic range is identified.     

Effect of a spoke surface error on a phase mask in a computational imaging system

The effect of a spoke surface error on a phase mask in a computational imaging system was analyzed by combining the similarity of the point spread function (PSF) and the peak signal-to-noise ratio (PSNR) of de-convoluted images. The spoke surface error was applied on a phase mask with different peak-to-valley (P-V) values with various numbers of spoke rings in simulation. The minimum requirement of PSF similarity will be determined by a given PSNR threshold, which relates the defocus aberration. Finally, it can be concluded that a low-spatial frequency surface error is critical for a cubic phase mask in a computational imaging system with lower P-V error.     

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.     

Cross-talk Noise and Storage Capacity of Holographic Memories with a LiNbO(3) Crystal in the Open-Circuit Condition

A solution to the Kukhtarev equations is obtained for a typical holographic memory system in which multiplexed holograms, including the effects owing to a nonuniform beam profile in the focal regions, are used. The various noise mechanisms and storage capacity are analyzed on the basis of this solution. The cross-talk noise of a typical 4f holographic memory configuration with defocus is compared with that of a phase mask. It is shown that the memory capacity and the signal-to-noise can be significantly improved by design of an optimal phase mask. The experimental results with defocus and an eight-level phase mask are presented.     

Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask

An approximate expression of the peak position of the point-spread function (PSF) of wavefront coding systems with a cubic phase mask (CPM) is derived and verified by simulation results. An approach called the nonaxial Strehl ratio (NASR) is used to evaluate the performance of wavefront coding systems with defocus aberrations. The characteristics of the NASR are investigated. The relationships between the NASR and the similarity of out-of-focus modulation transfer function (MTF) and recoverability of blurred encoding images are presented, and some useful guidelines are given for the optimization of phase mask parameters according to these relationships.     

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.     

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.     

Extended depth of field through wave-front coding

We designed an optical-digital system that delivers near-diffraction-limited imaging performance with a large depth of field. This system is the standard incoherent optical system modified by a phase mask with digital processing of the resulting intermediate image. The phase mask alters or codes the received incoherent wave front in such a way that the point-spread function and the optical transfer function do not change appreciably as a function of misfocus. Focus-independent digital filtering of the intermediate image is used to produce a combined optical-digital system that has a nearly diffraction limited point-spread function. This high-resolution extended depth of field is obtained through the expense of an increased dynamic range of the incoherent system. We use both the ambiguity function and the stationary-phase method to design these phase masks.     

Method of obtaining wavefront slope data from through-focus point spread function measurements

A method is described for analyzing point source imagery collected with various amounts of defocus to obtain wavefront slope data at the exit pupil of an imaging system. Integration of this slope data yields the system wavefront aberration function. The method is based on a geometric optics interpretation of intensity point spread function measurements in the caustic region near focus. Algorithm performance is examined through Monte Carlo simulations. Application of the method to segmented-aperture systems is also explored. The proposed method is used to generate initial wavefront estimates for an iterative phase retrieval algorithm, significantly improving the capture range over a blind phase retrieval approach when segment tilt errors are large.     

Signal-to-noise analysis of task-based imaging systems with defocus

We analyze the signal-to-noise ratio (SNR) of arbitrary imaging systems in the presence of defocus. The modulation transfer function (MTF) and the mean SNR are combined to calculate the spatial-frequency spectrum of the SNR (the spectral SNR). Computational imaging methods are used for extending the depth of field (DOF) of the system. The DOF of a task-specific imaging system is defined as the range of defocus that causes the spectral SNR to drop below a minimum value within a band of spatial frequencies of interest. We introduce the polar-SNR plot as a tool for visualizing the spectral SNR of defocused imaging systems with asymmetric pupil functions. As an example, we perform the analysis of an imaging system used for biometric iris recognition.     

Extended depth of field in hybrid imaging systems: Circular aperture

Abstract

A hybrid imaging system combines a modified optical imaging system and a digital post-processing step. A spatial-domain method is described to design a pupil phase plate to extend the depth of field of an incoherent hybrid imaging system with a circular aperture. This method is used to obtain a pupil phase plate to extend the depth of field, and is referred to as the EDF circular phase plate. By introducing an EDF circular phase plate at the exit pupil of a simulated diffraction-limited system and digitally processing the output of the detector, the depth of field is extended by approximately an order of magnitude more than the Hopkins defocus criterion.     

Pseudorandom phase masks for superresolution imaging from subpixel shifting

We present a method for overcoming the pixel-limited resolution of digital imagers. Our method combines optical point-spread function engineering with subpixel image shifting. We place an optimized pseudorandom phase mask in the aperture stop of a conventional imager and demonstrate the improved performance that can be achieved by combining multiple subpixel shifted images. Simulation results show that the pseudorandom phase-enhanced lens (PRPEL) imager achieves as much as 50% resolution improvement over a conventional multiframe imager. The PRPEL imager also enhances reconstruction root-mean-squared error by as much as 20%. We present experimental results that validate the predicted PRPEL imager performance.     

Defocused transfer function for a partially coherent microscope and application to phase retrieval

The defocused weak-object transfer function of a partially coherent bright-field microscope is calculated. For weak defocus, this can be expressed analytically. Use of this transfer function for phase restoration (quantitative phase retrieval) from images of weak mixed phase-amplitude objects is discussed.     

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.     

Quantitative characterization of the limiting resolution of a microscanning imager

A method to determine the limiting resolution of a microscanning imager is proposed. Specifically, both the sample-scene phase effects and aliasing effects due to microscanning are modeled in this method by combining the pixel transfer function and the squeeze modulation transfer function. Further, this model is used to calculate the amount of improvement from typical microscanning modes to the limiting resolution of the imager focusing on various blur factors. Analytical results show that the limiting resolution of the microscanning imager is closely related to microscanning modes. The amount of improvement from different microscanning modes to the limiting resolution is different and is closely associated with the fill factor and the blur factors. The conclusion obtained will be helpful in choosing the optimum microscanning mode according to the fill factor of the detector and system blur factors.     

High-diffractive-efficiency defocus grating for wavefront curvature sensing

The optimum phase defocus grating for wavefront curvature sensing is proposed. It features an equidistantly quantized, binary-phase-step defocus grating with a phase-step height of pi. The diffractive efficiency of the phase defocus grating is theoretically computed. The optical transfer function is obtained. The optimum phase defocus grating is fabricated. The high diffractive efficiencies of the +/-1 diffraction orders are verified experimentally, the average values of which are 38.08% and 40.36%, respectively.     

Combined amplitude and phase filters for increased tolerance to spherical aberration

Abstract

Analysis of the expression for Strehl ratio for a circularly symmetric pupil allows one to design complex filters that offer reduced sensitivity to spherical aberration. It is shown that filters that combine hyper-Gaussian amplitude transmittance with hyper-Gaussian phase modulation provide five-fold reduction in sensitivity to spherical aberration. Furthermore, this is achieved without the introduction of zeros into the modulation transfer function and deconvolution can restore the transfer function to that of a diffraction-limited imager. The performance of the derived combined amplitude and phase filter is illustrated through the variation of its axial intensity versus spherical aberration. This technique is applicable to imaging in the presence of significant amounts of spherical aberration as is encountered in, for example, microscopy.     

Single-lens single-image incoherent passive-ranging systems

We introduce a new system for single-lens single-image incoherent passive ranging. The only a priori object information this system requires is that the objects to be ranged must possess a low-pass spatial frequency spectrum. Physically, this system for passive ranging is a standard optical imaging system that is customized with a special-purpose optical mask or filter. Analytically, this optical mask customizes the transfer function of the optical system in such a way that objects form images that contain range-dependent information. This range-dependent information lies in the spatial spectrum nulls or zeros of the image.     

Modeling of through-focus aerial image with aberration and imaginary mask edge effects in optical lithography simulation

Aerial image through focus in the presence of aberrations and electromagnetic edge effects modeled by adding ±π/2 phase at pattern edges is expanded by a quadratic equation with respect to focus. The quadratic equation is expressed by four coefficients that are adequately independent of both mask layout and the variations in the optical setting in projection printing, thus saving the computation cost of the quadratic fit for each individual layout edge position in a new mask pattern or variation from a nominal optical setting. The error of this method is less than 1% for any typical integrated circuit features. This accuracy holds when the defocus is less than one Rayleigh unit (0.5λ/NA(2), where λ is a wavelength and NA is the numerical aperture) and the root mean square of the existing aberration is less than 0.02λ, which encompasses current lithography practice. More importantly, the method is a foundation for future first-cut accurate algebraic imaging models that have sufficient speed for assessing the desired or undesired changes in the through-focus images of millions of features as the optical system conditions change. These optical system changes occur naturally across the image field, and aberration levels are even programmed in tuning modern tooling to compensate for electromagnetic mask edge effects.