Cruz Meneses-fabian - Academia.edu (original) (raw)

Papers by Cruz Meneses-fabian

Research paper thumbnail of Interferometric sinograms of transparent objects with stepping wise shifted Ronchi ruling for optical tomography

Revista Mexicana de Fisica

Recibido el 21 de enero de 2005; aceptado el 28 de septiembre de 2005 An experimental set-up to a... more Recibido el 21 de enero de 2005; aceptado el 28 de septiembre de 2005 An experimental set-up to achieve tomographic images of slices belonging to transparent objects is presented. The proposed system is based on an interferometer which employs two windows in the object plane and a translating absorptive grating (Ronchi ruling) in the frequency plane. In the image plane, replicated windows can be brought to superposition by proper adjustment of the windows' spacing and the grating period. After adaptation of this interferometer to tomographic measurements, the phase of the projections result encoded as interference fringes, thereby forming a composite interference pattern in the plane of the corresponding sinogram (interfero-sinograms). Using phase-stepping techniques, four phase-shifted interfero-sinograms are found after proper translation of the grating. Then wrapped a and unwrapped phase is extracted as a conventional interferogram to obtain a sinogram to be subjected to a filtered backprojection algorithm for reconstruction. As experimental results, the reconstructions of some transparent samples are presented.

Research paper thumbnail of One-Shot Phase-Shifting Interferometry with Phase-Gratings and Modulation of Polarization Using N?4 Interferograms

Research paper thumbnail of Phase-Shifting Interferometry by Amplitude Modulation

Research paper thumbnail of Edge Enhancement Computed Tomography

In tomography, the probe must be able to cross the sample and carry the information of the struct... more In tomography, the probe must be able to cross the sample and carry the information of the structure to the outside. The action exerted upon the object due to the interaction with the medium is known as the projection or profile f  and the predominant physical property in

Research paper thumbnail of N-Shots 2N-Phase-Steps Binary Grating Interferometry

Research paper thumbnail of Non-quadrature amplitude modulation: a novel interferometric method for phase retrieval

European Journal of Physics, 2015

Phase-shifting interferometry (PSI) is a well-known technique used to calculate the phase of a re... more Phase-shifting interferometry (PSI) is a well-known technique used to calculate the phase of a resulting wavefront. This wavefront is commonly originated from the superposition of a known reference wavefront with a distorted wavefront, resulting in an interference pattern. In order to extract information about the resulting wavefront (which can be related to surfaces thicknesses, surface roughness, optical power, material homogeneity, temperature, index of refraction, just to mention a few), a set of interference patterns are generally changed in a known phase-step to form a resoluble system of equations. In PSI the contrast, visibility and phase variations of the resulting interference patterns are considered homogeneous and spatially constant, respectively. These conditions are hard to obtain experimentally, even though a carefully calibrated phase shifter and/or optical elements of high quality are used to carry out the experimental setup. In this manuscript, we present a novel and alternative method for phase extraction named phase-visibility modulating interferometry (PVMI), which introduces spatial variations of phase and visibility in the interferogram. The phase is computed from the measured intensities, and thus the interferograms can be normalized and their phase variations can be also known. The spatial variations are introduced by modulating in phase and amplitude a reference beam by non-quadrature amplitude modulation (NQAM), achieved by the sum of two beams out of phase within the range (0, 2Pi) and by the modulation of their amplitudes. NQAM is produced in this proposal by using: dark sheets as amplitude filters, neutral density filters and the diffraction orders of a grating, which considerably reduce the expenses generated when using common phase shifter devices such as piezoelectric transducers, besides that the amplitude filters need no calibration. In PVMI the spatial variations of phase and visibility do not represent any problem, however the phase difference between the reference beams has to remain temporally constant, which represent an experimental problem. For this reason, in order to overcome this inconvenient PVMI also has been implemented in a single shot. A theoretical analysis and experimental results will be shown for each implementation. The viability of these proposed experimental setups allows them to be easily replicated and used for researching purposes or for teaching special topics in optics for undergraduate or graduate studies in the field of physics.

Research paper thumbnail of Phase retrieval by Euclidean distance in self-calibrating generalized phase-shifting interferometry of three steps

Optics Express, 2015

This paper presents a novel algorithm for phase extraction based on the computation of the Euclid... more This paper presents a novel algorithm for phase extraction based on the computation of the Euclidean distance from a point to an ellipse. The idea consists in extracting the intensities from a data row or column in three interferograms to form points of intensity and then fitting them to an ellipse by the method of least squares. The Euclidean distance for each intensity point is computed to find a parametric phase whose value is associated to the object phase. The main advantage of the present method is to avoid the use of tangent function, reducing the error in the desired phase computation.

Research paper thumbnail of Generalized experimental phase extraction algorithm for speckle interferometry

Practical Holography XXIX: Materials and Applications, 2015

Research paper thumbnail of Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer

Journal of Optics, 2015

This work presents a method for generating carrier fringes and a non-conventional rotational shea... more This work presents a method for generating carrier fringes and a non-conventional rotational shear in a triangular cyclicpath interferometer, suppressing at the same time the presence of typical lateral and radial shearing. To carry out this method a 4f optical system is implemented into this cyclic interferometer. The most important contribution of this paper is the demonstration of linear dependence of the movable mirror displacement with the carrier frequency introduced and the obtaining of a non-conventional rotational shearing interferometer. Additionally, we think that one of its possible potential applications is the observation of the angular derivative of parallel projections of a phase object placed at the output plane, generating a great advantage in edge-enhancement optical tomography. In this paper, we develop a theoretical model and show experimental results.

Research paper thumbnail of Fully automated low-cost setup for fringe projection profilometry: Engineering and Lab Notes

Applied Optics, 2015

In this paper an alternative low-cost, easy-to-use, and fully automated profilometry setup is pro... more In this paper an alternative low-cost, easy-to-use, and fully automated profilometry setup is proposed. The setup is based on a phase-shifting fringe projection technique with four projected fringe parameters. It uses the well-known triangulation arrangement and low-cost electronic and image acquisition components such as a data acquisition board, a motor controller board, a printer rail, a CMOS webcam, and an LCD projector. The position of the camera, the generation of the fringe pattern, the acquisition of the images, and the calculation of the wrapped and unwrapped phase are all performed in LabVIEW. The setup is portable and can be perfectly adapted to be used in other profilometry techniques such as electronic speckle pattern interferometry and laser scanning profilometry.

Research paper thumbnail of <title>Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order 1/2</title>

ICO20: Optical Information Processing, 2006

Optical tomography ofphase objects could be carried out with optical derivatives at the phase det... more Optical tomography ofphase objects could be carried out with optical derivatives at the phase detecting stage. A particularly convenient related technique is the use of a fractional derivative of fractional order 1/2, which consists of a spatial filter with square-root amplitude transmittance. An advantage of this type of filter is that it can be implemented as a real filter when dealing with phase objects. Irradiance is proportional to the phase derivative of order one, and it can be obtained. If a tomographic detection system is equiped with such a filter, phase gradient of projections can be determined, and directional derivatives of phase slices can be displayed. Slice images resulting from an experimental tomographic system for parallel projections includes a spatial filter. with square-root amplitude transmittance made from a liquid crystal display operating in amplitude modulation are shown.

Research paper thumbnail of Modulation of Polarization for Phase Extraction in Holographic Interferometry with Two References

Heterodyne holographic interferometry allows high accuracy for phase-difference extraction betwee... more Heterodyne holographic interferometry allows high accuracy for phase-difference extraction between two wave fronts, especially when they are previously recorded in the same recording medium. In part, this is because the wave fronts can be affected by the recording process in a very similar way. The double reconstruction of a doubleexposure hologram with two independent references results in a two-beam holographic interferometer with an arm conveying a wave modulated in frequency when using heterodyne techniques. The heterodyne frequency has been usually introduced with a plane mirror attached to a piezo-electric stack driven with a suitable variable power supply. For holographic interferometry, however, less attention has been devoted to alternative phase retrieval variants as, for example, phase-shifting with modulation of polarization or Fourier methods. In this work, we propose and demonstrate the basic capabilities of modulation of polarization performing as a phase-shifting technique for holographic interferometry with two references in a phase-stepping scheme. Experimental results are provided.

Research paper thumbnail of Spatially inhomogeneous phase-modulating interferometry by non-quadrature amplitude modulation

8th Iberoamerican Optics Meeting and 11th Latin American Meeting on Optics, Lasers, and Applications, 2013

ABSTRACT A new method to generate inhomogeneous spatial phase variation and phase retrieval is pr... more ABSTRACT A new method to generate inhomogeneous spatial phase variation and phase retrieval is proposed. This method is based under the scheme of three beams forming two Mach-Zehnder interferometers in series, where one arm is considered as the probe beam, while the other two will compose the reference beam where a phase difference within the range of (0,π ) is kept and this remains spatially variable in opposition to the classical PSI method. The generation of phase-shifts is done by on-off nonquadrature amplitude modulation. Although the phase difference between the reference beams are spatially variable having as a consequence that the visibility and phase variations are inhomogeneous, it will be demonstrated that it is still possible to retrieve the phase of the object and also to measure the phase variation. An analytic discussion and experimental results will be shown to sustain the proposed method.

Research paper thumbnail of Lateral and rotational shearing phase-shifting interferometry with phase-grating multiplexing

Pacific Rim Conference on Lasers and Electro-Optics, CLEO - Technical Digest, 2009

Experimental results for the cases of n = 4,5,7 shearograms for one lateral shearing interferomet... more Experimental results for the cases of n = 4,5,7 shearograms for one lateral shearing interferometer and for one rotational shearing interferometer are shown. These results are obtained choosing one from several possible configurations of each system in only one-shot.

Research paper thumbnail of Phase-Shifting Interferometry by Amplitude Modulation

Research paper thumbnail of <title>Phase-shifting common-path interferometer with binary grating: application on optical tomography</title>

An experimental setup for phase extraction of 2-D phase distri-utions is presented. The system us... more An experimental setup for phase extraction of 2-D phase distri-utions is presented. The system uses a common-path interferometer consisting of two windows in the input plane and a translating grating as its pupil. In the output, interference of the fields associated with replicated windows is achie2ed -y a proper choice of the windows spacing with respect to the grating period. Because in this type of interferometer a grating is placed as a spatial filter, the phase changes which are needed for phase-shifting interferometry can -e easily performed with translations of the grating dri2en -y a linear actuator. Some experimental results as well as applications on Optical Tomography are shown.

Research paper thumbnail of <title>Phase shifting interferometry by modulation of the electric field module</title>

A new method of phase-shifting interferometry based on the wave amplitude modulation is proposed ... more A new method of phase-shifting interferometry based on the wave amplitude modulation is proposed and discussed. This proposal is based on the interference of three waves, where two waves attend as two reference waves and the other wave attends as a probe wave. Whereby, three interference terms are obtained, but because of a phase difference of 2 π between the two references is kept constant, one of the three terms will be dropped while the two remaining will be put in quadrature. Under these conditions the resulting pattern is mathematically modeled by an interferogram of two waves, where an additional phase is given by the amplitude variations of the reference waves. In this paper, both a theoretical model and some numerical simulations are presented.

Research paper thumbnail of Vector carrier frequency by interfering the kth harmonic of two rulings rotated polarly

Optics and Lasers in Engineering, 2015

This paper presents a method for introducing carrier fringes inclined at any angle into an interf... more This paper presents a method for introducing carrier fringes inclined at any angle into an interferogram. The setup is built on a 4f optical system consisting of two apertures in the input plane and a Ronchi ruling in the Fourier plane. Additionally, a Ronchi ruling rotated on polar direction is placed at each aperture and two passband filters are placed in the Fourier plane for filtering the kth harmonic of their spectra. We demonstrated that the magnitude and direction of the vector carrier frequency depend on the grating period at the input plane, the polar angles, and the kth harmonic, which gives this method the ability to modulate its magnitude only, or its direction only, or both in a wide range. The theoretical model and experimental results are shown in this paper.

Research paper thumbnail of Babinet’s principle for scalar complex objects in the far field

Journal of Optics, 2014

Babinet's principle is briefly reviewed, especially regarding the zeroth diffraction order of the... more Babinet's principle is briefly reviewed, especially regarding the zeroth diffraction order of the far field diffraction pattern associated with a given aperture. The pattern is basically described by the squared modulus of the Fourier transform of its amplitude distribution (scalar case). In this paper, complementary objects are defined with respect to complex values and not only with respect to unity in order to include phase objects and phase modulation. It is shown that the difference in complementary patterns can be sometimes a bright spot at the zero order location as is widely known, but also, it can be a gray spot or even a dark one. Conditions of occurrence for each case are given as well as some numerical and experimental examples.

Research paper thumbnail of Edge Enhancement of Phase Objects with Hilbert Transforms in Optical Tomography

The Hilbert transform (HT) is implemented in a 4f Fourier-transform imaging optical system by pla... more The Hilbert transform (HT) is implemented in a 4f Fourier-transform imaging optical system by placing a phase step filter of π radians in the frequency plane.

Research paper thumbnail of Interferometric sinograms of transparent objects with stepping wise shifted Ronchi ruling for optical tomography

Revista Mexicana de Fisica

Recibido el 21 de enero de 2005; aceptado el 28 de septiembre de 2005 An experimental set-up to a... more Recibido el 21 de enero de 2005; aceptado el 28 de septiembre de 2005 An experimental set-up to achieve tomographic images of slices belonging to transparent objects is presented. The proposed system is based on an interferometer which employs two windows in the object plane and a translating absorptive grating (Ronchi ruling) in the frequency plane. In the image plane, replicated windows can be brought to superposition by proper adjustment of the windows' spacing and the grating period. After adaptation of this interferometer to tomographic measurements, the phase of the projections result encoded as interference fringes, thereby forming a composite interference pattern in the plane of the corresponding sinogram (interfero-sinograms). Using phase-stepping techniques, four phase-shifted interfero-sinograms are found after proper translation of the grating. Then wrapped a and unwrapped phase is extracted as a conventional interferogram to obtain a sinogram to be subjected to a filtered backprojection algorithm for reconstruction. As experimental results, the reconstructions of some transparent samples are presented.

Research paper thumbnail of One-Shot Phase-Shifting Interferometry with Phase-Gratings and Modulation of Polarization Using N?4 Interferograms

Research paper thumbnail of Phase-Shifting Interferometry by Amplitude Modulation

Research paper thumbnail of Edge Enhancement Computed Tomography

In tomography, the probe must be able to cross the sample and carry the information of the struct... more In tomography, the probe must be able to cross the sample and carry the information of the structure to the outside. The action exerted upon the object due to the interaction with the medium is known as the projection or profile f  and the predominant physical property in

Research paper thumbnail of N-Shots 2N-Phase-Steps Binary Grating Interferometry

Research paper thumbnail of Non-quadrature amplitude modulation: a novel interferometric method for phase retrieval

European Journal of Physics, 2015

Phase-shifting interferometry (PSI) is a well-known technique used to calculate the phase of a re... more Phase-shifting interferometry (PSI) is a well-known technique used to calculate the phase of a resulting wavefront. This wavefront is commonly originated from the superposition of a known reference wavefront with a distorted wavefront, resulting in an interference pattern. In order to extract information about the resulting wavefront (which can be related to surfaces thicknesses, surface roughness, optical power, material homogeneity, temperature, index of refraction, just to mention a few), a set of interference patterns are generally changed in a known phase-step to form a resoluble system of equations. In PSI the contrast, visibility and phase variations of the resulting interference patterns are considered homogeneous and spatially constant, respectively. These conditions are hard to obtain experimentally, even though a carefully calibrated phase shifter and/or optical elements of high quality are used to carry out the experimental setup. In this manuscript, we present a novel and alternative method for phase extraction named phase-visibility modulating interferometry (PVMI), which introduces spatial variations of phase and visibility in the interferogram. The phase is computed from the measured intensities, and thus the interferograms can be normalized and their phase variations can be also known. The spatial variations are introduced by modulating in phase and amplitude a reference beam by non-quadrature amplitude modulation (NQAM), achieved by the sum of two beams out of phase within the range (0, 2Pi) and by the modulation of their amplitudes. NQAM is produced in this proposal by using: dark sheets as amplitude filters, neutral density filters and the diffraction orders of a grating, which considerably reduce the expenses generated when using common phase shifter devices such as piezoelectric transducers, besides that the amplitude filters need no calibration. In PVMI the spatial variations of phase and visibility do not represent any problem, however the phase difference between the reference beams has to remain temporally constant, which represent an experimental problem. For this reason, in order to overcome this inconvenient PVMI also has been implemented in a single shot. A theoretical analysis and experimental results will be shown for each implementation. The viability of these proposed experimental setups allows them to be easily replicated and used for researching purposes or for teaching special topics in optics for undergraduate or graduate studies in the field of physics.

Research paper thumbnail of Phase retrieval by Euclidean distance in self-calibrating generalized phase-shifting interferometry of three steps

Optics Express, 2015

This paper presents a novel algorithm for phase extraction based on the computation of the Euclid... more This paper presents a novel algorithm for phase extraction based on the computation of the Euclidean distance from a point to an ellipse. The idea consists in extracting the intensities from a data row or column in three interferograms to form points of intensity and then fitting them to an ellipse by the method of least squares. The Euclidean distance for each intensity point is computed to find a parametric phase whose value is associated to the object phase. The main advantage of the present method is to avoid the use of tangent function, reducing the error in the desired phase computation.

Research paper thumbnail of Generalized experimental phase extraction algorithm for speckle interferometry

Practical Holography XXIX: Materials and Applications, 2015

Research paper thumbnail of Carrier fringes and a non-conventional rotational shear in a triangular cyclic-path interferometer

Journal of Optics, 2015

This work presents a method for generating carrier fringes and a non-conventional rotational shea... more This work presents a method for generating carrier fringes and a non-conventional rotational shear in a triangular cyclicpath interferometer, suppressing at the same time the presence of typical lateral and radial shearing. To carry out this method a 4f optical system is implemented into this cyclic interferometer. The most important contribution of this paper is the demonstration of linear dependence of the movable mirror displacement with the carrier frequency introduced and the obtaining of a non-conventional rotational shearing interferometer. Additionally, we think that one of its possible potential applications is the observation of the angular derivative of parallel projections of a phase object placed at the output plane, generating a great advantage in edge-enhancement optical tomography. In this paper, we develop a theoretical model and show experimental results.

Research paper thumbnail of Fully automated low-cost setup for fringe projection profilometry: Engineering and Lab Notes

Applied Optics, 2015

In this paper an alternative low-cost, easy-to-use, and fully automated profilometry setup is pro... more In this paper an alternative low-cost, easy-to-use, and fully automated profilometry setup is proposed. The setup is based on a phase-shifting fringe projection technique with four projected fringe parameters. It uses the well-known triangulation arrangement and low-cost electronic and image acquisition components such as a data acquisition board, a motor controller board, a printer rail, a CMOS webcam, and an LCD projector. The position of the camera, the generation of the fringe pattern, the acquisition of the images, and the calculation of the wrapped and unwrapped phase are all performed in LabVIEW. The setup is portable and can be perfectly adapted to be used in other profilometry techniques such as electronic speckle pattern interferometry and laser scanning profilometry.

Research paper thumbnail of <title>Tomographic directional derivative of phase objects slices using 1-D derivative spatial filtering of fractional order 1/2</title>

ICO20: Optical Information Processing, 2006

Optical tomography ofphase objects could be carried out with optical derivatives at the phase det... more Optical tomography ofphase objects could be carried out with optical derivatives at the phase detecting stage. A particularly convenient related technique is the use of a fractional derivative of fractional order 1/2, which consists of a spatial filter with square-root amplitude transmittance. An advantage of this type of filter is that it can be implemented as a real filter when dealing with phase objects. Irradiance is proportional to the phase derivative of order one, and it can be obtained. If a tomographic detection system is equiped with such a filter, phase gradient of projections can be determined, and directional derivatives of phase slices can be displayed. Slice images resulting from an experimental tomographic system for parallel projections includes a spatial filter. with square-root amplitude transmittance made from a liquid crystal display operating in amplitude modulation are shown.

Research paper thumbnail of Modulation of Polarization for Phase Extraction in Holographic Interferometry with Two References

Heterodyne holographic interferometry allows high accuracy for phase-difference extraction betwee... more Heterodyne holographic interferometry allows high accuracy for phase-difference extraction between two wave fronts, especially when they are previously recorded in the same recording medium. In part, this is because the wave fronts can be affected by the recording process in a very similar way. The double reconstruction of a doubleexposure hologram with two independent references results in a two-beam holographic interferometer with an arm conveying a wave modulated in frequency when using heterodyne techniques. The heterodyne frequency has been usually introduced with a plane mirror attached to a piezo-electric stack driven with a suitable variable power supply. For holographic interferometry, however, less attention has been devoted to alternative phase retrieval variants as, for example, phase-shifting with modulation of polarization or Fourier methods. In this work, we propose and demonstrate the basic capabilities of modulation of polarization performing as a phase-shifting technique for holographic interferometry with two references in a phase-stepping scheme. Experimental results are provided.

Research paper thumbnail of Spatially inhomogeneous phase-modulating interferometry by non-quadrature amplitude modulation

8th Iberoamerican Optics Meeting and 11th Latin American Meeting on Optics, Lasers, and Applications, 2013

ABSTRACT A new method to generate inhomogeneous spatial phase variation and phase retrieval is pr... more ABSTRACT A new method to generate inhomogeneous spatial phase variation and phase retrieval is proposed. This method is based under the scheme of three beams forming two Mach-Zehnder interferometers in series, where one arm is considered as the probe beam, while the other two will compose the reference beam where a phase difference within the range of (0,π ) is kept and this remains spatially variable in opposition to the classical PSI method. The generation of phase-shifts is done by on-off nonquadrature amplitude modulation. Although the phase difference between the reference beams are spatially variable having as a consequence that the visibility and phase variations are inhomogeneous, it will be demonstrated that it is still possible to retrieve the phase of the object and also to measure the phase variation. An analytic discussion and experimental results will be shown to sustain the proposed method.

Research paper thumbnail of Lateral and rotational shearing phase-shifting interferometry with phase-grating multiplexing

Pacific Rim Conference on Lasers and Electro-Optics, CLEO - Technical Digest, 2009

Experimental results for the cases of n = 4,5,7 shearograms for one lateral shearing interferomet... more Experimental results for the cases of n = 4,5,7 shearograms for one lateral shearing interferometer and for one rotational shearing interferometer are shown. These results are obtained choosing one from several possible configurations of each system in only one-shot.

Research paper thumbnail of Phase-Shifting Interferometry by Amplitude Modulation

Research paper thumbnail of <title>Phase-shifting common-path interferometer with binary grating: application on optical tomography</title>

An experimental setup for phase extraction of 2-D phase distri-utions is presented. The system us... more An experimental setup for phase extraction of 2-D phase distri-utions is presented. The system uses a common-path interferometer consisting of two windows in the input plane and a translating grating as its pupil. In the output, interference of the fields associated with replicated windows is achie2ed -y a proper choice of the windows spacing with respect to the grating period. Because in this type of interferometer a grating is placed as a spatial filter, the phase changes which are needed for phase-shifting interferometry can -e easily performed with translations of the grating dri2en -y a linear actuator. Some experimental results as well as applications on Optical Tomography are shown.

Research paper thumbnail of <title>Phase shifting interferometry by modulation of the electric field module</title>

A new method of phase-shifting interferometry based on the wave amplitude modulation is proposed ... more A new method of phase-shifting interferometry based on the wave amplitude modulation is proposed and discussed. This proposal is based on the interference of three waves, where two waves attend as two reference waves and the other wave attends as a probe wave. Whereby, three interference terms are obtained, but because of a phase difference of 2 π between the two references is kept constant, one of the three terms will be dropped while the two remaining will be put in quadrature. Under these conditions the resulting pattern is mathematically modeled by an interferogram of two waves, where an additional phase is given by the amplitude variations of the reference waves. In this paper, both a theoretical model and some numerical simulations are presented.

Research paper thumbnail of Vector carrier frequency by interfering the kth harmonic of two rulings rotated polarly

Optics and Lasers in Engineering, 2015

This paper presents a method for introducing carrier fringes inclined at any angle into an interf... more This paper presents a method for introducing carrier fringes inclined at any angle into an interferogram. The setup is built on a 4f optical system consisting of two apertures in the input plane and a Ronchi ruling in the Fourier plane. Additionally, a Ronchi ruling rotated on polar direction is placed at each aperture and two passband filters are placed in the Fourier plane for filtering the kth harmonic of their spectra. We demonstrated that the magnitude and direction of the vector carrier frequency depend on the grating period at the input plane, the polar angles, and the kth harmonic, which gives this method the ability to modulate its magnitude only, or its direction only, or both in a wide range. The theoretical model and experimental results are shown in this paper.

Research paper thumbnail of Babinet’s principle for scalar complex objects in the far field

Journal of Optics, 2014

Babinet's principle is briefly reviewed, especially regarding the zeroth diffraction order of the... more Babinet's principle is briefly reviewed, especially regarding the zeroth diffraction order of the far field diffraction pattern associated with a given aperture. The pattern is basically described by the squared modulus of the Fourier transform of its amplitude distribution (scalar case). In this paper, complementary objects are defined with respect to complex values and not only with respect to unity in order to include phase objects and phase modulation. It is shown that the difference in complementary patterns can be sometimes a bright spot at the zero order location as is widely known, but also, it can be a gray spot or even a dark one. Conditions of occurrence for each case are given as well as some numerical and experimental examples.

Research paper thumbnail of Edge Enhancement of Phase Objects with Hilbert Transforms in Optical Tomography

The Hilbert transform (HT) is implemented in a 4f Fourier-transform imaging optical system by pla... more The Hilbert transform (HT) is implemented in a 4f Fourier-transform imaging optical system by placing a phase step filter of π radians in the frequency plane.