Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination

Analytical expressions for the spatial distribution of the spectral density and intensity of projection images are derived for a broad class of object transmission functions in the case of partially coherent Schell-model-type incident illumination. The expressions are linear with respect to the phase distribution in the transmitted wave. Associated methods for phase retrieval are discussed with the emphasis on a technique that allows simultaneous “automatic” phase retrieval and deconvolution of projection images of homogeneous objects by means of mutual cancellation at a specified defocus distance of the Fresnel diffraction effects and the image blurring due to the point-spread function of the imaging system.     

Methods of Fourier optics in digital holographic microscopy

In this paper, processing methods of Fourier optics implemented in a digital holographic microscopy system are presented. The proposed methodology is based on the possibility of the digital holography in carrying out the whole reconstruction of the recorded wave front and consequently, the determination of the phase and intensity distribution in any arbitrary plane located between the object and the recording plane. In this way, in digital holographic microscopy the field produced by the objective lens can be reconstructed along its propagation, allowing the reconstruction of the back focal plane of the lens, so that the complex amplitudes of the Fraunhofer diffraction, or equivalently the Fourier transform, of the light distribution across the object can be known. The manipulation of Fourier transform plane makes possible the design of digital methods of optical processing and image analysis. The proposed method has a great practical utility and represents a powerful tool in image analysis and data processing. The theoretical aspects of the method are presented, and its validity has been demonstrated using computer generated holograms and images simulations of microscopic objects.     

Phase contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems

We treat the problems of phase-contrast image formation, deterministic phase retrieval and aberration balancing, in the imaging of weak objects using two-dimensional shift-invariant linear imaging systems. Three classes of model sample are considered: weak phase objects, weak phase-amplitude objects and single-material weak phase-amplitude objects. For each class of sample we show how the various aberration coefficients, which characterise a given imaging system, contribute to the structure of the associated phase-contrast image. The corresponding inverse problem, of obtaining a closed-form expression for the input wave-field given one or more aberrated phase-contrast images of the same, is then examined. Two sample applications are considered: analyser-crystal phase-contrast imaging of weak objects using hard X-rays, and Zernike-type phase-contrast imaging. We close with a discussion of how coherent and incoherent aberrations may be “balanced” against one another, briefly mentioning the applications of this idea to both “deblur by defocus” and proximity-corrected X-ray lithography.     

Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift

An iterative algorithm to extract the arbitrary unknown phase shift in two-frame phase-shifting interferometry and then reconstruct the complex object wave is proposed. In combination with the least square principle and some calculation formulae we developed, this algorithm allows us to find the value of unknown phase shift by using only two interferograms without additional knowledge or measurement. Computer simulations have shown that this algorithm works well for both the smooth and diffusing objects to a very high accuracy over a wide range of the phase shift from 0.4 to 2.5 rad.     

Phase-coded aperture for optical imaging

For the optical spectrum region, we describe a novel phase-coded aperture imaging system that can be used in a computational imaging camera. The optical design includes a phase-only screen followed by a detector array. A specific diffraction pattern forms at the detector array when the wavefront from a point source object passes through the phase screen. Since diffraction effects cannot be ignored in the optical regime, an iterative phase retrieval method is used to calculate the phase coded screen. Correlation type processing can be applied for the image recovery. Computer simulation results are presented to illustrate the excellent imaging performance of this camera.     

Propagation characteristics of laser beams with amplitude modulations and phase fluctuations in apertured fractional Fourier transforming system

The apertured fractional Fourier transforming system is introduced and applied to treat the propagation of Gaussian beams with amplitude modulations and phase fluctuations. Based on the treatment that a rectangular function can be expanded into an approximate sum of complex Gaussian functions with finite numbers, the analytical expressions for the mutual intensity distribution of laser beams with amplitude modulations and phase fluctuations passing through the apertured fractional Fourier transforming system are obtained. Some numerical simulations are illustrated for their propagation properties.     

Fraunhofer holography of small particles

The reconstruction of an in-line Fraunhofer (far-field) hologram is re-examined, and an analytical solution of the intensity distribution throughout the entire reconstructed image is presented. The solution bridges the gap between previously documented solutions which are limited to the plane-of-focus image and distant out-of-focus images. The analysis is motivated by a particle velocity measurement technique which attempts to distinguish the focussed image by photographic thresholding. The general methodology is presented for objects of one- and two-dimensional cross-section. Specific results are presented for the single exposure holograms of a long wire and a small particle of circular cross-section. The one-dimensional solution is verified experimentally. The results show precisely how the Fresnel diffraction term creates peaks in the intensity distribution, both upstream and downstream of the focussed image. This characteristic limits the resolution of methods which use thresholding as a means of distinguishing focussed images from their out-of-focus neighbors.     

Necessity analyses of phase masks in joint fractional Fourier transform correlator

In this paper, the mathematical expressions and the particularities of the joint fractional Fourier transform correlator (JFRTC) are detailedly analyzed. In a JFRTC, the distance between the two fractional correlation peaks can be adjusted flexibly by shifting the fractional orders. However, the intensity of the fractional correlation peaks decreases rapidly as the first fractional order p₁ drops. The former property is very useful in a practical recognition system, but the latter one seriously reduces the recognition sensitivity of a JFRTC. For this reason, phase masks should be added at the input plane of the JFRTC to enhance the intensities of the fractional correlation peaks and effectively overcome the default. Compared with the existing JFRTC with single phase modulation, a JFRTC with double phase modulation is suggested to more flexibly improve the performance of a JFRTC.     

Novel Ghost Imaging Method for a Pure Phase Object

We propose a novel ghost imaging scheme which is especially served to a pure phase object. A spatially incoherent beam is mixed with a coherent beam of the same frequency field by a beamsplitter. Then we perform the ghost imaging scheme using the mixed beam. Our theoretical result shows that this approach is capable of reconstructing a pure phase object in joint-intensity measurement. The visibility of the images is also analysed for two pure phase objects, an optical wedge and a phase grating.     

Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition

When the Hilbert transform method is used to recover the phase distribution in temporal speckle pattern interferometry, the influence of the fluctuations of the bias and the modulation intensities on the calculated phase must be driven heuristically. In this paper we show that the Empirical Mode Decomposition method can overcome these drawbacks and consequently introduces an improvement in the evaluation of the phase distribution. An example is used to illustrate the phase measurement improvement that can be obtained by the application of the proposed approach.     

Real-time recognition of microscopic phase objects

A method of image enhancement and real-time input of 3-D, microscopic phase objects into a coherent optical pattern recognition system is described. The method consists of directing a low-power laser beam into a microscope objective to produce a real, magnified, coherent image of the specimen under test. The image plane is followed by two successive Fourier transform (FT) planes. In the first FT plane, low and high frequency spatial filters, one of which is photographically produced, are used as pre-processing filters to enhance the image quality. The enhanced signal is imaged from the first FT plane to the second FT plane which contains a matched spatial filter used for specimen identification. The system does not require an expensive incoherent-to-coherent light transducer and in addition, is capable of utilizing both phase and amplitude information from 3-D objects. Examples of results are given.     

Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain

A novel digital image watermarking system based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform (DCT) domain is proposed. The original hidden image is first encrypted into two phase masks. Then the cosine and sine functions of one of the phase masks are introduced as a watermark to be embedded into an enlarged host image in the DCT domain. By extracting the watermark of the enlarged superposed image and decryption we can retrieve the hidden image. The feasibility of this method and its robustness against some attacks, such as occlusion, noise attacks, quantization have been verified by computer simulations. This approach can avoid the cross-talk noise due to direct information superposition and enhance the imperceptibility of hidden data.     

Optimal Truncation of Element Beam in a Coherent Fiber Laser Array

The beam quality of a coherent fiber laser array often suffers from the low fill factor of the Gaussian laser array. One simple and effective approach to improve the fill factor is to truncate the array element properly. An analytical expression for far-field intensity distribution of a truncated coherent fiber laser array is derived. Optimal truncation of the element beam in different coherent fiber laser arrays is obtained by using energy encircled in the far-field central-lobe as the beam quality criterion. By optimal truncation, energy encircled in the central-lobe can be 97% compared with the ideal case. The shift in optimal truncation parameter in the case of phase noise is also analyzed.     

Modified temporal-phase-unwrapping method for measuring in real time the out-of-plane displacements of the tympanic membrane of Mongolian Gerbil

A technique for measuring in real-time continuous out-of-plane displacements of delicate objects is proposed, and demonstrated on the tympanic membrane of Mongolian Gerbil. The technique is based on the combination of two methods: the spatial phase shifting (SPS) and the modified temporal phase unwrapping (TPU). The combination allows to obtain, in several steps, the phase values of the points that undergo out-of-plane displacement as the object is deformed. The technique reduces the frame acquisition time of the standard TPU used in moiré interferometry by a factor of 4, which is important to diminish post-mortem artifacts during in-vitro experiments and to reduce motion artifacts in in-vivo tests. The proposed technique is robust against problems associated with the temporal phase-shifting method, such as nonlinear phase shift and noise. The advantages and disadvantages are discussed.     

Transformation of a hollow Gaussian beam into a ring-shaped beam using a phase aperture

Propagation of a hollow Gaussian beam diffracted by a circular phase aperture is studied without making the paraxial approximation. The analytical expression of the intensity of the apertured hollow Gaussian beam is presented in the far field. The influences of the truncation parameter and the order of hollow Gaussian beam on the intensity distributions are discussed. It is shown that a circular ?-phase aperture can be used to transform a hollow Gaussian beam into a ring-shaped beam in the far field with the appropriate parameters.     

The focus of light – theoretical calculation and experimental tomographic reconstruction

We present numerical calculations on the field distribution in the focus of an optical system with high numerical aperture. The diffraction integrals which are based on the Debye approximation are derived and evaluated for a radially polarized input field with a doughnut-shaped intensity distribution. It is shown that this mode focusses down to a spot size significantly smaller as compared to the case of linear polarization. An experimental setup to measure the three-dimensional intensity distribution in the focal region is presented, which is based on the knife-edge method and on tomographic reconstruction. Received: 14 July 2000 / Published online: 30 November 2000     

Real-time nondestructive imaging with THz waves

We present a real-time imaging measurement in the terahertz (THz) frequency region. The dynamic subtraction technique is used to reduce long-term optical background drift. The reflective images of two targets, a Nikon camera’s lens cap and a plastic toy gun, are obtained. For the lens cap, the image data were processed to be false-color images. For the toy gun, we show that even under an optically opaque canvas bag, a clear terahertz image is obtained. It is shown that terahertz real-time imaging can be used to nondestructively detect concealed objects.     

The filtering characteristics of simple grating optical low-pass filters

We studied the characteristics of a two-dimensional grating optical low-pass filter (GOLF) theoretically and experimentally. The modulation transfer function (MTF) of an optical system that consists of a lens and a GOLF is theoretically derived by taking all orders of diffracted beams into consideration. The MTFs of a two-phase chess-board-type GOLF and a three-phase GOLF were calculated for various phase differences and compared with that of a birefringent low-pass filter (BLF). The three-phase GOLF with nine center beams of equal strength removes most moiré fringes, but the resolution degradation is severe compared to the BLF. The two-phase GOLF with a phase difference of 180°, which is similar to the BLF in term of beam distribution, has a medium characteristic somewhere between those of the three-phase GOLF and the BLF. Samples of two GOLFs are made and experimented on by attaching them to a digital camera. The experimental result coincides with the theoretical development. Received: 31 October 2001 / Revised version: 4 March 2002 / Published online: 2 May 2002     

On the origins of decoherence and extinction contrast in phase-contrast imaging

A theoretical formalism describing the formation of images in a linear shift invariant X-ray optical system is derived within the wave-optical theory. It is applicable to a non-crystalline object consisting of two types of features, with the characteristic sizes which are respectively not smaller and much smaller than the resolution of the imaging system. This formalism is then applied to two phase-contrast imaging techniques, the propagation-based and analyser-based imaging. The obtained formulae for the intensity distribution in the image well explain the “decoherence effect” which is observed in the former technique and the “extinction contrast” which is a characteristic of the latter technique. This formalism is shown to be in good agreement with the results of the accurate numerical simulations, using rigorous wave-optical theory, of the propagation-based and analyser-based phase-contrast images of the model objects.     

Intensity and phase distribution in deep Fresnel diffraction region generated by multi-pinhole aperture

The Kirchhoff diffraction theory is applied to the multi-pinhole aperture diffraction screens, and the intensity, the zero-contour of the real and imaginary parts of complex amplitude and the phase distribution in deep Fresnel diffraction region are simulated. It is found that the number of bright spots, the zero-contours of the real and imaginary parts of complex amplitude and the phase singularities are all related to number of pinholes in diffraction screens. The brightness of bright spots in center of each pattern would become larger with increasing number of pinholes. In addition, there are many lines, on which the intensity value is close to 0.